Теория вероятностей и математическая статистика

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Конспект лекции по дисциплине «Теория вероятностей и математическая статистика», pdf

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An ∈ A. ]\}42 ¯¡M=03:C87&0=N&B!C57&>: =8Z=2;62;B37dN&03B3[¦| C5:"[\7&<3D^] ¯ @ $p jUpr* M 03:D3T3v &N Z=82;"0=NªM3:~46B3:A©2;C87&> Ω ]B!N&_8[\>UN&2~6[¦Ω,|CU,Z=vPw=N"D3B3[6<%ΩC5:"[\7&<@=6<] P ¯w3<=C5Z=:">UNd@1vB3T3=<@^]:"M30=24!28UZ=28B3B!Ndσ@ B=N U N"2;6Nd@q>2803:A@37&B=:C57"`Q>S]U"v;4!2~6B=N"_8[\>UN&7&`k>2803:A@37"B3:C87&B3[6/M30=:C57"0=N&B=C87&>":U6¦]2;CZ!&Z=2;7&>":Q@ 0U@32879CZ!24!v > ª<=6 N&TC5<3:U6N"6-J , 4!Z3@?>UC528| A ∈ A xzB328:7&03<3=N"7&25Z!`"B3:UC57&` P 1 P(A) ≥ 0 , x B=:"0=6<30=:">N"B3B3:UC57&` P 2 P(Ω) = 1 , ]!2;C5Z=< A ∩ B = ∅ x NA434<37&<3>B3:C87&` k 3 P(A + B) = P(A) + P(B) , L ® O C = Z µ < 8 C " : " \ [ " 7 < @ 9 > 3 M U : C = Z 5 2 4!:">UN&7&28Z=`"B3:UC57&< {A } M=:"M=N"03B3:9B32;C8:">U62;C57"B3[F]7&: 4 A n=1 n=1 ◦ & ◦ & !' ◦ ◦ n & n P ∞ X ∞ X An = P(An ) . s®>:"D=C87&>Nª>"2;03:A@37&B3:UC57&B=:"D{1vUB=T33<3":D=C57">": Q >"[\M=:&Z=B@32;7"C}@tB=NO>2803:A@37&B=:C57"B3:6 M=03:C87&0=N&B!C57&>2 C1>2803:A@37&B=:C57"Q` >y+"]7&:q;:">":0U@37~]w=7&:qC5>":D=C57">": Q <=62;287q62;C87&:qM3:w37&<{B=N&>2803B=:"29x M^, B^,J, (Ω, U, P) k & k ] ! B V 2&: 3$3%-,.83/ %4 %-$3/ 6D,.83/ :(,.83/!% 3$3/ :(,.$ 18&:;,.4!/ .1vUC57&`XM303:UC57&0!N&B=C87&>":5= Z!2;628B=7AN"03B3[¦| C8:"[\7&<3D ¯B32{":"Z=282{w32~6C8w3287"B3:"2%6B3:A©2;C57">":=] @ NdZ=;280=N CZ!vUw!N&D3B3[¦|qC5:"[\7&<=D U ¯C5<=C87&2;6Nk>C528|M=:~46B3:A©ª2~C57&>k6ΩB=:A©ª2;C87&>N Ω 2803:A@37&B=:C57"B=Nd@q1vB3T33<@:σM303254!25Z3@L@ 287C}@?_~NA4N"B3<32;6 >"2;03:A@37&B3:UC57&< P(ω) TN&©ª4:"8:9<=C}|U:~4N ω ∈ Ω ]3v;4!:>&Z=2;7&>":0@@{C5::"7&B3:S2;B3 P P(ω) = 1 , M 03:"D=TN Ω, U, P B=N"_8[\>UN&287C}@c4<=C5T30=287&B3[6µ>"280=:A@37&B3:C87&B3[6{M30=:C57"0=N&B=C87&>":U6¦,dK®2;03:A@37&B3:UC57&`cC5:"[\7&<@ A ∈ Ω ω∈Ω _;Nd4N&2;7"C}@0!N&>"2;B=C57">":6 P(A) = X P(ω). GFNC57&B3[6 CZ=vw=N"2;6O4<=C5T30=287&B3:8:9>"2;03:A@37&B3:UC57&B=:"8:ªM303:UC57&0=N"B=C57">NS@3>&Z3@32;7"C}@7dN&T?B=N&_;[\>N"2;6:"2cTZ!NC8C52803:A@37"B3:C87&B3:"2\M=03:C87&0=N&B!C57&>:=]&T:"}4)N1>"280=:A@37&B3:C87&C825|9<=C}|:~4:">G0=N&>B3[ 4!03v¦4!03v;v",¬GZ!N"C;C5<3w32~C5TNd@ 6:~4!28Z=`1>"280=:A@37&B3:C87&B3:8:1M303:UC57&0=N"B=C57">N1<=C8M3:&Z=`_8v287"C}@S7":"}4N=]&T:"}4N =5Z!2;628B=7AN"03B3[\2C8:""[\7"<@0!N&>"B=:">":_;6:A©D@ B3[F, E ":"_;B=NAw3<=6 C5<=6>:&Z=:U6 |A| w3<=CZ!: =5Z=2~628B37&:>6B3:A©ª2~C57&>UN A , M N&T¡TN&T 1 = P P(ω) = P(ω)|Ω| ] 7&: ω∈A ω∈Ω P(A) = |A| . |Ω| j 1Upo^! !rp'3!pr#pr WYC87AN"B3:">"Z=28B=B3[\D9>kT:"B32;w3B3:6O6B3:A©2;C57">"2\M3:0@4:"T9B=N&_;[\>N"287"C}@M328032~C57dN&B=:">"T:D^,&%B3:A©2;C57">":FC\_;Nd4N&BF@ B3[6 M3:"0U@4!T:U6B=N"_8[\>N"287"C}@vUM=:"0U@4!:"w=28B3B3[6 6B3:A©ª2~C57&>:6¦, ¬1:B328w3B=:"2vUM=:"0U@4!:"w=28B3B3:2ª6B3:A©2;C87&>":?B=N&_;[\>N"287"C}@µ0=N"_;62;S28B=<32;6¦, G1<=C5Z=:?0=N&_~628S2;B3<3D A :"Q P2;6N >k q6B3:A©ª2~C57&>2G<=_ n =5Z=2~628B37":">90=N&>B3: k n Akn = 2;6N n! . (n − k)! ¬1:B328w3B=[\2®M3:~4)6B3:A©ª2~C57&>UN\T:"B32;w3B3:";:c6B3:A©2;C87&>N B=N&_;[\>N?>c7"C}@SC5:w3287dN"B3<@=6<^, G1<=C5Z=:GC8:"w32;7dN&B3<3Dª:"NP @ >q6B3:A©2;C87&>"21<3_ n =5Z=2~628B37&:>q0=N&>B3: m Cnm = n! m!(n − m)! # #!('3 !1' .1vC57"`µ<=6282;7"C}@tv03B=NC :~4!<=B=N&T:">[6UN&B3B=[¦|1w=<=CZ!N6< 1, 2, . . . , N ,&.10=24!M3:"Z=:A©qN<=6¦]dw37&:1N&03[ C®B=:6280!N"6< 1, 2, . . . , M "28Z=:";: 3>287dN3]Y:UC57dNdZ!`"B3[\2?¯ w=2803B3:8:!,uKL[\":"0=TNX28_%>:"_8>0=N&ª28B3<3@ C5:UC57&:<37/>M3:UCZ=254!:">UN&7&28Z=`"B=:6<=_8>&Z!28w32;B3<3< n ªN"03:">!]^B32>":"_;>"0=N"ªN&@R<|X:"0=N&7&B=:=,pK=87":6 CZ!vUw!N&2_~N%M303:UC57&0=N"B=C57">"*: =8Z=2;62;B37dN&03B3[¦|C8:""[\7"<3D 6:A©qB3: M303<=B@37&`{6B3:A©ª2~C57&>:>C825|%vM3:"0U@4!:w328B=B3[¦|%B=N&:"03:> (ω , ω , . . . , ω ) w3<=C825Z ω , 1 6 ω 6 N ]B=Ω2ª0=N">"B3[¦| 4!03v4!0=vU;v", E S282w3<=C5Z=:%<=C}|U:~4!:">3> =;7&:6 CZ!vUw!N&290=N&>B3: A ¯µw3<=C5Z=v0!N&_;62;S2;B3<3DO<3_ N =8Z=2;62;B37&:>?M=: ,^KL[\w3<=C5Z=<=6 >2803:A@37"B3:C87&`?C8:"[\7&<@ ]^C8:C87&:A@3S2;8:>{7&:U6¦]w37&:%C50=24!"Z=28w=28B3B3[¦| ªN"03:">?<=6282;7"C}@ n 03:>"B3: m 25Z=[¦|, G1<=C5Z=: "Z!N&;:"M303<3@37&B3[¦B|k<=C}|U:~4!:">\:M3032425Z3@32;7"C}@STN&TªM303:"<=_8>"254!28B=<32uw3 >"[\":0=N TU::"04!<3B!N&7/<3_µ:"ª288:/<3|T:"Z=<3w32~C57&>UN ]®B!N/T:"7&:03[\26[E0=N"_;62;C87&<=6 B=:6280!N/"28Z=[¦| N&03:>=] ¯ m w3<=C5Z!N90=N"_5Z=<3w=B3[¦|{C8M3:C8:":">0=N&_~62;C57"<37&`25nZ=[\2FN&03[ B=N m 62~C57< A ¯w3 0=N"_;62;C87&<37&`qw=2803B3[\2FN&03[¨B=N:C87AN">"ª<32;C}@ n − m 62;C57A,=K <37&:";2"]3M=:&Z=vw=N"2;6 % % 1 2 n n N k k m n−m N −M P(Bm ) = n−m Cnm Am M AN −M . AnN .1:&Z!`"_8v"@=C5`ªTU:U6"<3B!N&7&:03B3[6<{1:"0=6v~Z!N6<^]3>"2;03:A@37&B3:UC57&` P(Bm ) = P(Bm ) m n−m CM CN −M . n CN 6:A©qB3:9M30324)C57dN&><37&`9>9>"<4!2 m M uO C5Z="Z=28w32;B3<@XªN"03:">!])N{7":&Z=`TU:{C5:UC57dN&>{>[\":"0=T3<^])7&:%>"[\0=N&7"` m 25Z=[¦| N&03:>ª<3_ n ªN"03:">96:A©qB3: C C C5M3:UC5:N"6<M303<%:"S2~6T:&Z!<3w32;C87&>"21<=C}|U:~4!:">!]0!N&>"B=:6 C ,=.1:N=87&:U6v =87&:U6tC5Z=vw=N&21>"2;03:A@37&B3:UC57&`qC8:""[\7"<@ B "v;4!2;7ª0!N&>"B!N m M n−m N −M n N m P(Bm ) = m n−m CM CN −M . n CN M N&T3<=6X:"0=N"_8:U6¦]d>kM303<3>24!2;B3B3:U6RM=03<=6280=2>"[\:"03T3<928_>:"_8>0=N&ª28B3<3@ =5Z!2;628B=7AN"03B3[6<C8:""[\7"<@=6<6:A©qB3: 8C w3<37dN&7&`9TN&T?vM3:"0U@4!:w328B=B3[\2"]7dN&T?<{B32FvM3:"0U@4!:w328B=B3[\2c>[\":03T3<^, .1vC87&`S<=62;287"C}@?7dNk©2cv03B=N=]<3_GTU:7&:"0=:"DM=:CZ!24!:>N&7"25Z=`B3:k<3_;>&Z=2;TN"287"C}@ M3:?:~4!B3:U6v{ªN&0=v"]1<3TC5<30=v"287C}@µB3:62;0ªN"0=N3]N?C8N6¡ªN"0>":_8>0=N&N&287C}@?:"0=N"7&B3:>?v03B3v,^K=87&:U6 CZ=vw=N"2 M30=:C57"0=N&B=C87&>"3: =5Z!2;628B=7AN"03B3[¦|RC8:"[\7&<3DRC8:C87&:"<37?<3_q>28T37&:03:"> (ω , ω , . . . , ω ) ]T:":0"4<3B=N&7"[T:"7&:03[¦| 1 6 ,3 K =;7&:6 CZ=vw=N"2 ω 6N |Ω| = N , N>"2;03:A@37&B3:UC57&`9C5:"[\7&<3@ A 0=N&>B=N " % %# 1 j 2 n n m P(Am ) = Cnm m M n−m M m (N − M )n−m m M = C 1 − . n Nn N N O®CZ={ªN"03:">!]N%B32M=:CZ!24!:>N&7"25Z=`B3:C87&`=]>%T:"7":"03:D =87&&Z=2;TN?> 7C}@^]I=8Z=2;62;B37dN&03B=[6"2;T37&:03[ (n , . . . , n ) ]}4!2 n + · · · + n = n ,=!4!2~C5` n w3&Z=2;w328B3B=[¦|ªN"03:">9C1B3:62;03:6 k ,=K =;7&:6 CZ!vUw!N&21>"2;03:A@37&B3:UC57&`qC5:"[\7&<@ A 0=N&>B=N 1 N 1 N k m P(Am ) = n−m m CM +m−1 CN −M +n−m−1 . n CN +n−1 £®7":"790328_;v~Z=`d7dN&7ªZ=28;TU:9>[\>":~4!<37C}@<3_1Z=2~66[ 11'z "!$#%&'&(*)+!-,. # # !-,. r Cn+r−1 r1 + · · · + rn = r, rk > 0, / i !1'0Up#ljpjr .1vUC57&` U, P) >2803:A@37&B=:C57"B3:"29M303:UC57&0!N&B=C87&>":!]>?T:7&:"03:U6=8Z=2;62;B37dN&03B=[6&Z3@F>c7"C}@ >UC528>:"_~6:A©9B=[\(Ω, 2M3:C5Z=24:">N"7&25Z!`"B3:UC57&C5ω28|c<=C}|U:~4!:">!]5C5k:~4!2;0U©N&ª<| k 254!<3B3<3,;P(Ω) KL2803:A@3=7"B3:C87&"B=N p (1−p) A ] M3:N=87&:U6v>"2;03:A@37&B3:UC57&`9C5:"[\7&<3@ A 0=N">"B=N k r k=0 n−k k k n n−k k k n−k k P(Ak ) = Cnk pk (1 − p)n−k . bFN"":0µ>"2;03:A@37&B3:UC57&2;D B=N&_;[\>N"287"C}@µ<3B3:62803:Q@ @37&B=:C57"B=Nd@{C}|2;6NS¯C}|2~6:"D&SL28C03B3pv~Z3(1Z!<^−, p) 3 GU#rpm&#r#µ(U#&r .1vUC57&` ¯S:&Z!NC57&`128>TZ=<4:">NGM30=:C57"0=N&B=C87&>N C\T:"B32;w3B3[6 @ 62;03B3[6O:"NP^2;6:U6¦,Us®:"[\7&<@=6<9B=N?@ _;:">"2~6<=_;6280=<=Ω6[\2®M=:~46B3:A©ª2~C57&>UN Ω ,~N NdZ=;28"0=vGC8:""[\R7"<3DFM303<=62~6 σ@ NdnZ=;28"0=v B :"0328Z=28>UC5T3<|GM3:~46B3:A©2;C87&> ,=N>2803:A@37&B=:C57"`9C8:"[\7&<@ A ∈ B M303<=62~6 Ω k k n n−k n k=0 n P(A) = |A| , |Ω| a }4!2 :"":_8B=NAw!N&287 @ 62803B=[\Dt:"NP^2;6 6B=:A©ª2;C87&>N V , M :}4NR7&03:"D=TN (Ω, B, P) @3>&Z3@32;7"C}@>"2;03:A@37&B3:UC57&B=[6 M303:UC5|V7&0!|N&B=C87&>":U6¦, n bFNMZ=:UC5T:C87&<^])0=N"C8w3280=w328B3B=:"DµM=N"0=NdZ=Z=25Z=`B3[6<M=0@=6[6<]B=Nd|U:~4=@L@ ! # ! S"2;03:A@37&B3:UC57&` C5:"[\7&<@ A = {<= Z!NSM=28032~C528w=2879TN&T3v > @ B3<3v;4!`9<3_1M=N&0=N&Z3Z=28Z=`"B3[¦|M30U@=6[¦| } , P(A) E :"_8B!NAw3<=6?w3280=28_ 0=N"C;C57&:A@3B3<=2u:"7cC5280=24!<3B=[<3 Z=[X4: &Z!<©N"D3S2;DkM30U@=6:"D^] ¯1:UC57&03[\Dªv8:&Z625©4!v <3 Z=:D<{M32;03M32;B4!<3T3v~Z3@30=:y6 T?M!N&0=N&Z3Z=25Z!`"B3[6/M=0@=6[6¦,¬1:":0"4<3B=N&7"[ (x, y) :"M303254!25xZ=@F> 79M3:&Z!:A©ª28B=<32c<= Z=[ :"7"B3:C8<37&25Z!`"B3:/M=N"0=NdZ=Z=25Z=`B3[¦|tM30U@=6[¦| <¡v84:">&Z!287&>:"0UF@ > 7XvC5Z=:"><@=6 0 6 x 6 π/2, 0 6 y 6 `/2 ,YbFN MZ=:UC5T:C87&< (x, y) :"B3<9:"0=N"_8v > 71M30U@=6:"v;:&Z=`B3<3T Ω , 1 Z!NGM3280=2;C52;w3287F:~4B3vS<3_\M!N&0=N&Z3Z=25Z!`"B3[¦|M30U@=6[¦|)]"2;CZ!< < v;4!:">"Z=287">":"0UF@ >c7ªB32;0=N&>28B=C87&>"v x y 06y6 FC5T:U6Nd@>2803:A@37&B=:C57"`0!N&>"B!N ! |A| P(A) = = |Ω| :&/!4!8 ` cos x . 2 π/2 R ` 2 cos xdx a/2 · π/2 = 2` . aπ %4!%-$&/067,98&/!:;,92&8&% !4!23:;23)+/!:;, dj=(#o9µ(U#&r .1vUC57&`{M303< <=C8M3[\7dN&B3<3@|OC5:"[\7&<3@ A ] B < AB M303:<3_8:kZ=":UC57&`w=N"C87&:"7 N N < N ≈ P(AB) ≈ P(A), ≈ P(B) N N N < P(B) > 0 ]7&::7&B3:UC5<37&28Z=`"B!Nd@µw=NC57&:7AN N /N C8:""[\7"<@ A M=03<vUCZ=:>"<3<^]!w37&:?M303:<3_8:kZ=:?C5:"[\7&<32 B 7dN&T©21vC57":"D3w3<=>N A B A AB B AB AB B NAB NAB /N P(AB) = ≈ . NB NB /N P(B) ,)WYCZ!:">"B=:"D>2803:A@37"B3:C87&`Q> $p j#r .1vUC57&` w37&:qM=03:"<3_;:"SZ=:C8:"[\7&<32 B ]3B!N&_8:>"P(B) 2~6/:"7&B=>:"S0 2;B3<32 P(A|B) = P(A|B) C5:"[\7&<3@ A M303"<=<^] P(AB) . P(B) YW CZ!:">"B!Nd@>2803:A@37"B3:C87&` P(A|B) :":"_;B=NAw=N&2;7"C}@7dN&T©2 P (A) , OuC5Z=<C5Z=vw=N&D=B3:"2 C8:"[\7&<32 <3_ B32;T:"7&:03:";:F>"280=:A@37&B3:C87&B3:8:FM303:UC57&0!N&B=C87&>N 1<3TC5<=03:">UN&B3:!] 7&:kvUCZ=:>"B=N&@>2803:A@37"B3:C87&` P (A)BTN"T91vB3T=3<@CZ!vUw!N&D3B3:8:FC5:"[\7&<3@ A :"M30=24!28Z3@3287FΩ,B3U,:>":P)2\>"280=:A@37&B3:C87&B3:2 M303:UC57&0!N&B=C87&>": Ω, U, P ) , =1'z x 7&2;:"032~6Nv6B=:A©ª28B=<@!, ) '&( '& A , . . . , A ' ,"! $#&% ' B B B P(A1 . . . An−1 ) > 0 /(' *),+ , 1 n P(A1 . . . An ) = P(A1 )PA1 (A2 )PA1 A2 (A3 ) . . . PA1 ...An−1 (An ) . QS:TN"_8[\>UN&287C}@6287&:~4!:U6 6N"7&2;6N"7&<3w32~C5T:"D?<3B4vUT=3<3<^, - !#Glj'%$#=j#!n¡(U#&r (1) $p jUpr ¢^ s®<=C57&2~6vkC5:"[\7&<3D :B3<M=:"M=N"03B3:9B32;C8:">U62;C57"B3[ < A1 , A2 , . . . , An "v;4!2;6µB=N&_;[\>N&7"` T:"B32;w3B3[6%0!N&_8<328B=<32;6¦]A2;C5Z=< A1 + A 2 + · · · + A n = Ω . !#1'µ h x k:0=6v~Z!NFM3:"Z=B3:Dq>"280=:A@37&B3:C87& 0 P(Ak )P(B|Ak ) , # '& (2) k=1 B = BΩ = BA1 + BA2 + · · · + BAn , M3:N=87&:U6v P(B) = .103 0 # ' ##' P(Ak )P(B|Ak ) P(Ak |B) = Pn . i=1 P(Ai )P(B|Ai ) *! , ,' # ( '& .1:ª7&2;:"032~62Fv6B3:A©ª2;B3<@ / P(Ak B) = P(Ak )P(B|Ak ) = P(B)P(Ak |B) , M3:N=87&:U6v P(Ak |B) = P(Ak )P(B|Ak ) P(B) E C87AN&Z=:C8`SM30=<=628B3<=7&`T{_8B=N628B=N"7&25ZK> P(B) 1:0=6v~Z=vM3:"Z=B3:"D?>"2;03:A@37&B3:UC57&<, bFN"_8:>"2;6 C5:"[\7&<@ 8<3M3:7&28_~N"6<^,.1vC87&`C5:"[\7&<32 0328_;v~Z=`d7dN&79B328T:7&:"03:8:7=;TC8M3280=<=628B37dN=,!KL2803:Q@ @37&B=:C57"< P(A ) ¯38= 7": ,.! A ! # >2803:A@37&B=:C57"<µ8<3M=:"7&2;_"]>"[\Bw3<=C5Z3@32;6[\2k4!:M=03:">24!2;B3<@µ:"M=[\7AN=,=WYC5Z=:">B3[\2 >2803:A@37&B=:C57"< P(A |B) ¯ 8= 7": ,.' # ! ! # >2803:A@37"B3:C87&<^]>"[\w=<=CZ3@32~6[\21M3:C5Z=21M303:">24!2;B3<@?:"M3[\7dN3, k k k U'(!r&rGoqµ" =o9r# .1vUC57&`\>2803:A@37"B3:C87&`\C5:"[\7&<@ B32Y0!N&>"B!N¦B3v~ZC>S,"OuC57&2~C57&>28B3B=:cC5w=<37dN&7&`!]5w37&:GC5:"[\7&<32 A B32u_;N&><=C5<=7 :7C8:""[\7"<@ B ]32;C5Z=< P(A|B) = P(A)P(B) ,COuC5Z=<{M303<&=87&:U6t< P(A) > 0 ]37&: P(B|A) = P(AB) P(B)P(A|B) P(B)P(A) = = = P(B) , P(A) P(A) P(A) A7 , 2", B B=2_;N">"<=C8<37{:"7 A , 1_q7"28:"0=2;6[ v6B3:A©ª2;B3<@X7&:"}4N{C5Z=24!v287A]w=7&:?4=Z3@/B32;_;N&><=C5"2;B=C57">": P(AB) = P(A)P(B) ,£®7&:GM303<=>":~4!<37cTC5Z=24vI>cS2~6vk:"M30=24!28Z=28B3 B32;_;N">"<=C8<=A6:C87&B<^, $p j#r s®:"[\7&<@ A < B=N"_8[\>UN?> 7C}@B328_~N&><=C5<=6[6<]32;C5Z=28B=C87&>":B32F>"[\M=:&Z=B@32;7"C}@^]7&:C8:""[\7"<@v842;6B!N&_8[\>UN&7&`9_~N&>""280=:A@37&B3:C87&B3:DR6:~4!28Z="<=C8<=6:C87&`C8:"[\7&<3DOB32ªvUC57dN&B!N&>&Z!<3>N"287"C}@^]N M3:UC57&v~Z=<30=v"287C}@?B=N:UC5B3:>"2GM303<3w3<=B3B3:"D{B32;_;N">"<=C8<=6:C87&<0=2;NdZ!`"B3[¦|M=03:":"0=N"_8:">!, $p j#r s®:"[\7&<@ A , A , . . . A B=N&_;[\>N?c> 7"C}@B32;_;N&><=C5 "[¦| 1 6 ] ]=>"[\M3:"Z=B@F> 7C}@0=N">"28B!C57&>UN i B3:6 CZ!vUw!N&2FC5:"[\7&<@?B=N&_;[\>NQ> 7"C}@_~N&><=C5<=6[6<, 1_1M3:M=N&03B=:"D?B328_~N&>"<=C5<=6:UC57&<, >[\"<30=N"287"C}@/:~4!B=:=]M303<3w=2;6 >UC52?<=C}|:~4![ 0!N&>"B=:">":_;6:A©qB3[ ! # ! .1vUC57&`O<3_?w3<=C528Z 7O>"2;03:A@37&/ B3:UC57&` 1/4 ,pKL>"254!2;62,w3283,032;_ 5,A30= {>"[\0=N&B=B3:"2cw3<=CZ!:4!25Z=<=7"C}@?B!N k} ,F2;8T:µ><4!287"`=]w37&: C5:"[\7&<3@ A , A , A M3:M=N&0=B3:9B328_~N&>""9C8:">:"T3vM3B3:UC57&<]U7dN"T?TN"T k 2 3 5 P(A2 ) = P(A3 ) = P(A5 ) = 1/2, P(A2 A3 ) = P(A2 A5 ) = P(A3 A5 ) = 1/4 1_FC5:>62~C57&B3:DB=28_;N">"<=C8<=6:C87&< < P(A2 A3 A5 ) = 1/4 6= 1/8 . P(A1 A2 . . . An ) = P(A1 )P(A2 ) . . . P(An ) . B32kC5Z=24!v287B328_~N&><=C5<=6:UC57&<, ! # ! / N"C;C86:7&03<=6°M30=:C57"0=N&B=C87&>": Ω ]#C8:C57":A@3S282q<=_H"µvUM=:"0U@4!:"w=28B3B3[¦|OM=N&0 (i, j) ]#}4!2 i, j = , S®v;4!2~6¨C8w3<37dN&7"`=]^w37":%>C529M=N"03[y0=N&>B3:">:"_;6:A©qB3[F, M :}4N4=Z3@/C5:"[\7&<=D A = {(i, j) : j = 1, 2, 5} ] 1, . . . , 6 < <=6A2 >c762;C57":C5:":7&B3:S28B=<@ B = {(i, j) : j = 4, 5, 6} C = {(i, j) : i + j = 9} P(AB) = b1:9>97&:9©ª21>"0=2;6#@ 1 1 6= = P(A)P(B), 6 4 P(AC) = 1 1 6= = P(A)P(C), 36 18 P(BC) = 1 1 6= = P(B)P(C). 12 18 P(ABC) = 1 = P(A)P(B)P(C). 36 11' h^ 1 '& A B # ., $#2'& '& Ā B ' ,"! # # ,. '& A B '& A B ' ,"!. # # ,. A A = ∅ # '& A + A 2 # ., / ◦ C57">"2 ◦ 1 2 1 2 1 2 B $p j#r ^³ ¥\Z!82803[°x σ@ NdZ=;28"0=[ C5:"[\7&<3D U , U , . . . , U ⊂ U >1>"280=:A@37&B3:C87&B3:U6µM303:UC57&0!N&BL@ B=N"_8[\>NQ> "7 C}@B328_~N&>"c"[¦| A ∈ U ] i = 1, n ]=>"[\M3:"Z=B@3287C}@0=N&>28B=C87&>": (Ω, U, P) 1 2 i n i P(A1 A2 . . . An ) = P(A1 )P(A2 ) . . . P(An ) . +~g D % !412&:(2&) % 2&: , 18 2 6 .1:~4µ<=C8M3[\7dN&B3<=2;6M3:B3<=6N&2~6B328T:7&:"03[\D ;= TC5M32;03<=628B=7~])<=C}|U:~4N"6<{T:"7":"03:8:CZ!v©9N&7C5Z=vw=N&D3B=[\2ªC5:Q@ [\7&<@^,.1:N=87&:U6v6:~4!25Z!`Q> <=C5M=[\7AN"B3<@O@3>&Z3@32;7"C}@R>2803:A@37"B3:C87&B3:"29M30=:C57"0=N&B=C87&>":!,^.1vC87&`{M303:">24!2;B3[ n <=CR@ M3[\7dN"B3<3D^]7&:q2~C57&`4)N&B3[¨>"280=:A@37&B3:C87&B3[\21M303:UC57&0!N&B=C87&>N Ω 1 , U1 , P 1 , . . . , Ω n , Un , P n . OuC5Z=< =87&<=C5<=6[F]7": @ N&Z=82;"03[4!:&Z=©9B=[[\7&`{B328_~N&>":"03<=7&`F7&:&Z!`"T:k0=NC8C86N"7&03<3>UNd@ σ@ N&Z=82;"03[ σ U TN"T σ@ M=:~4NdZ=;2803[ σ@ NdZ!82803[B328T:7&:"03:8:1:"S2;8: >2803:A@37&B=:C57"B3:";:LM=03:C87&0=N&B!C57&>UN3, M N&T:"2u>"280=:A@37&B3:C87&B3:2M303:C87&0=N"B=C57&>:>C828}4N\6:A©qB3: M3:C87&03:"<=7&`=,~.103:~4!25ZN&2;6 =;7&:9>9C5Z=vw=N&2]T:"}4N>"2;03:A@37&B3:UC57&B=[\2cM30=:C57"0=N&B=C87&>N Ω , U , P T:"B32;w3B3[F, ¯ T:"B32;w3B3[\2R>"280=:A@37&B3:C87&B3[\2µM=03:C87&0=N&B!C57&>UN3] .1vUC57&` ] ¯N&Z=82;"0=N/>UC528| M3:~4)6B3:A©ª2~C57&>O<3Ω_ ,ΩU ] , PP (A) = P P(ω ) ,®.1:"Z=:A©q<=6 Ω = Ω × · · · ×ΩΩ =={ω{ω} =U(ω , . . . , ω )} ]Y}42 ] ¯NdZ=;28"0!NS>UC525|M3:~46B3:A©2;C57">6B3:A©2;C87&>N Ω ] ω ∈Ω U k k k k k k k k k k k 1 k k k n 1 n ωk ∈A k P(ω) = P1 (ω1 ) · · · · · Pn (ωn ), P(A) = X P(ω) . LK >"254!28B=B=Nd@>2803:A@37"B3:C87&` P B=N&_;[\>N"287"C}@µM30U@=6[6¨M303:<3_8>24!2;B3<32;6 >"280=:A@37&B3:C87&28D P "254!28B=<32NdZ!8280^,®K M3:UC57&0=:"28B=B3:6 P × ··· × P >2803:A@37&B=:C57"B3:6 M303:UC57&0=N"B=C57">"2ªC8U:"=[\7&U<32k×><4· ·N · A× =U A × · · · × A "v;4!2;6B!N&_8[\>UN&7&`qM=0@=6:v8:&Z!`"B3<3T:U6¦, 1_ :M3032425Z=2;B3<@>2803:A@37"B3:C87&"2;03:A@37&B3:UC57&`M30U@=6:"v8:"Z=`"B=<3TN A 0=N">"B=N ω∈A k 1 n 1 n 1 P(A) = .1vUC57&` X P(ω) = X n P1 (ω1 ) · · · · · X Pn (ωn ) = :":"_;B=NAw=N&2;7kM3:~4N&Z=82;"03vq>UC528|M30U@=6:"v8:"Z=`"B=<3T:">S><4N ω1 ∈A1 ω∈A Uk0 ωn ∈An n Y Pk (Ak ) . k=1 A0k = Ω1 × · · · × Ωk−1 × Ak × Ωk+1 × · · · × Ωn ∈ Uk0 . µ25©4!v C8:"[\7&<@=6< ∈ U < A ∈ U vUC57dN&B=N">&Z=<=>N&2;7"C}@2;C87&2;C87&>"2;B3B3[\D<3_;:6:"0!1<3_;6 A ∼ A ] C8:A|0!N&B@F>cS<3D>"2;03:A@37&B3:UC57&< AP(A ,¦.1:Q=87":6v C8:"[\7&<@ A <3_>2803:A@37"B3:C87&B3:";:XM303:C87&0=N"B=C57&>UN ) = P (A ) ; v ! 4 ; 2 6 " : " 7 A : ª © 4 ; 2 5 C " 7 & > 3 Z 3 @ & 7 ` 1 C = < 8 _ : 6 : = 1 3 B [6<%C5:"[\7&<3@=6< A <3_ (Ω, U, P) , M N&T%TN&T A = A × · · · × (Ω , U , P ) T ]37":q<3_F:"M30=24!28Z=28B3<3@>"2;03:A@37&B3:UC57&<=C5"<=C8<=6[¦|k<=C5M=[\7AN"B3<3D9CLI <=C}|U:~4N"6GTNd©4!:U6%<=C5M=[\7AN"B3<3<^, .1vUC57&` >"2;03:A@37&B3:UC57&` M303:UC57&0=N"B=C57">N (Ω , U , P ) 7dN&T:">UN3]Lw37&: Ω = {0, 1} ] A = {∅, {0}, {1}, Ω } ] ] ] , P{0} = p P{1} = q = 1 − p i = 1, . . . , n K M30U@=6:6 M303:"<=_8>"254!28B=<3< (Ω, U, P) =5Z=2~628B37dN"03B3[\D°<=C}|:~4 ω ∈ Ω <=6282;7>"<4 ω = (ω , . . . , ω ) ] ,®KL2;03:A@37&B3:UC57&` P{ω} 0!N&>"B!N Q p (1 − p) , FC}|:~4 {1} "v;4!2;6 B!N&_8[\>UN&7&` ) # ]uM30=:"7&:"M3:"Z=:A©qB3[\D9<=C}|:~4 {0} @ #) # , E :"_;B=NAw3<=6Rw3280=28_ B = {ω : ω + · · · + ω = k} C8:""[\7"<32"]UC5:C87&:A@3S2;2 >7&:6¦]w=7&:9M303< B32;_;N&><=C59C}|2~62 SL280=B3v~Z3Z=B3: k vC5M=25|:>=,.1:C8TU:"Z=`"T=v 4=Z=@CZ >c":";:9<=C}|n:~4N ω ∈ B "280=:A@37&B3:C87&28D ++ =B N&_;[\>N&2;7"C}@ , ( !-,..! # +# # # , =jrp!Gr'j!' U1' K¡TN&©ª4:6/B32;_;N">"<=C8<=6:6X<=C8M3[\7dN&B3<=<>:"_;6:A©qB3:UC57&`ªM3:A@3>&Z!28B3<@:~4B3:";:S<3_ M3:M=N&0=B3:ªB=2;C5:>62~C57&B3[¦| <=C}|U:~4!:">!,u.1vC87&`>"2;03:A@37&B3:UC57&B=:"2%M303:UC57&0!N&B=C87&>": (Ω , U , P ), i = 1, . . . , n ]L7dN&T:"r>:=]Yw=7&: Ω = {1, . . . , r} ] @Y>"2;03:A@37&B3:UC57&!] ]< C8:C87&:"<379<3_F>UC528|M3:~46B3:A©2;C57"><=_ , p ,...,p K¨M30U@=6:6 M303:<3_8>24!2;B3<3< (Ω,p U,+P)· · ·=5+Z=2~p628=B37d1N&0=B3:"U2SC8:"[\7&<32 ω ∈ Ω <=62;287><4 ω = (ω Ω, . . . , ω ) ]=}42 ¯B3:U6280{<=C}|U:~4NkM30=< i@ :6t<=C5M3[\7dN"B3<3<^,=.1:"Z!N&5N"2;6 P{ω} = p p . . . p , ω KL>"254!2;6¡C8:"[\7&<32 B > B328_~N&>"B3: n k@ |?<=C}|U:~4!:">!] ={ n M , " N ? T T " N T = 4 = Z @ CZ >c":8: ω ∈ B n + n + · · · + n = n} {Cnk pk q n−k }nk=0 i 1 r 1 r i i i i i 1 i ω1 ω2 n1 ,...,nr 1 2 n ωn k r n1 ,...,nr p{ω} = r Y pnk k <T:"Z=<3w32~C57&>:!=5Z=2~628B37":">S> B 0=N&>B3: U] 7&: , bFN&:"0q>"2;03:A@37&B3:Q@ C57"28D {P(B )} ] n + · · · + n = n ]B=N"_8[\>N"287"C}@ P(B , ( ) = !-,.! # p+ # #.. . p# , E M3<=C8N"B3B=N&@C}|2~6N C r N"287"C}@ , ( , k=1 n1 ,...,nr n1 ...nr 1 r n! n1 !...nr ! n1 ...nr n! n1 !...nr ! n1 1 nr r + 1< 7A! 8 % 4!% & 232 8 7/183%38 6 : %0) Fb NAw=B32;6 Cc:M303254!25Z=2;B3<@%CZ=vw=N"D3B3:D>25Z=<3w=<3B3[F, $p jUpr ´ N"C;C86:"7"03<=6 T:"B32;w3B3:"2q>2803:A@37"B3:C87&B3:"2qM=03:C87&0=N&B!C57&>: U, P) ,G1<=C5Z=:">v > 1vB3T3=°:"7D=5Z=2~628B37dN"03B3:";:qC5:"[\7&<@ ξ = ξ(ω) ] ω ∈ Ω "v;4!2~6tB!N&_8[\>UN&7&`9CZ!vUw!N&D3B3:D(Ω, >25Z=<=w3<3B3:D^, ! # ! / G1ªM303< n C}|2~62 SL2;03B3v~Z3Z=< CZ!vUw!N&D3B=N&@>25Z=<3w=<3B=N3, ! # ! .1vUC57&`\> vU0=B32u<=62;287"C}@ N&03:>=]5<=_YB3<| "28Z=[¦|]8:C87dNdZ=`B3[\2w3280=B3[\2",~.1: C}|2~62Y>"[\:"0=N 28_1>":_8>"0!N&S2;B3<@/ <=_Gv03B=[¨<3_;>&Z=2;T?N > 7C}@ Nn ªN&0=:">=L, G1<=C5Z=:9M<3_8>"Z=28w=28B3B3[¦|"25Z![¦|ªN"03:">9>9>"[\:"03T2G:"Q P2;6N @3>&Z3@32;7"C}@?C5Z=vw=N&D3B=:"D?>"25Z!<3w3<3B3:D^, n ! # ! / s TNd©4![6 C5:"[\7&<32~6 A ∈ U 6:A©qB3:qC8>&@3_;N"7&`9CZ=vw=N"D3B3v >°>"25Z!<3w3<3B3v IA = IA (ω) = 1, ω ∈ A 0, ω ∈ / A, B=N"_8[\>UN&2;6v > <3B4!<=TN"7&:"0=:6tC8:"[\7&<@ A , F2;8T:M3] 03:">2803<=7&`qCZ=25] 4!v > S<=2kC5>":D=, C57">Nª<3B4!<3TN"7&:"03:>CJ ] I =0 I =1 I O®CZ=<%C5:"[\7&<3@ =AI ,I. . . ,IA =M3:"1M!−N&03IB3:9B32~C5:>62;C87&B3[F]7&: Ω ∅ AB A B A Ā 1 n n IP Ak = n X I Ak . ¯_;B=NAw32;B3<@^]3M30=<3B3<=6N"2;6[\2SC5Z=vw=N&D3B=:"D{>"28Z=<3w3<3B=:"D $p jUpr ! .1vC87&` ,ξ G1280=28_ A :":"_8B!NAw3<=6%CZ!vUw!N&D3B3:2®C8x:""[\<7"x<32 A< ·=· · {ω< x∈ Ω ] ξ(ω) = x } , M :"}4)N α = {A } 2;C57"` !-,. ## ,N&_;"<32;B3<32 α M=:"03:A©4N"2879NdZ=;28"0=vC5:"[\7&<3D U ]F=5Z=2~628B37dN6"Z3@F> 7"C}@%C5:"[\7&<3@ Ω k=1 k=1 1 i 2 n i i ξ i n i i=1 ξ ξ }4!2 B @ZC> :"2Yw=<=CZ=:>":2u6B3:A©ª2~C57&>:=,;N"_8<328B3<=2 α <ªN&Z=82;"03v U v;4!2;6{B=N"_8[\>UN&7&` ! + # ) %-, * # % ξ , 1> ":2®C5Z=vw=N&D3B=:"2®C8:"[\7&<32 {ξ ∈ B} M30324)C57dN&><=6: >c>"<4!2¦C5v66 P A ]A}4!2LC5v66<303:>N&B=<32 >24!2;7"C}@M3:97&2~6 i ]4!Z3@?T:7&:"03[¦| x ∈ B , b12;7&03v;4!B3:9><4!287"`=]3w37":CZ=vw=N"D3B=Nd@>"25Z!<3w3<3B=N ξ M303254C57dN&>"Z3@3287C}@>9>"<42 {ξ ∈ B} = {ω : ξ(ω) ∈ B}, ξ ξ i i i ξ = ξ(ω) = k X xi IAi (ω). $p jUpr 3h$,"! !-,.! # + # # CZ!vUw!N&D3B3:Dt>25Z=<3w=<3B3[ v;4!2;6 B=N&_;[\>N&7"`O1vB3T3= w3":";:S6B3:A©2;C57">N B : B → P{ξ ∈ B} ,3N"T:"B0=NC5M30=24!28Z=28B3<3@ ξ :M3032425Z3@32;7"C}@_8B!NAw328B3<3@=6< x , . . . , x < >"2;03:A@37&B3:UC57~@=6< P{ξ = x } = p , i = 1, n, P p = 1 ,uN"TU:Bt0=NC5M30=24!28Z=28B3<3@ v;4!:"B3:X_;N"M3<=C8N"7&`O> 1:0=6217AN"&Z=<=31[ J ,;,8, x ξ x x ,;,8, p P p p i=1 1 i i i i 1 2 n 1 2 n n +~H K 7&28:03<32803:A@37&B=:C57"<ªw=NC57&:S8:">:"0U@37G:kCZ=vw=N"D3B3:"D9>25Z=<3w=<3B32C\_~N&T:"B3:U6µ0=NC5M30=24!28Z=28B3<3@^]"B32\vTN&_;[-@ >N&@O>2803:A@37"B3:C87&B3:";:%M303:UC57&0=N"B=C57">N (Ω, U, P) ,p.103254!M3:&ZN&5N&2;7"C}@^]^w37":µ7dN&T:"2q>2803:A@37&B=:C57"B3:"2qM30=:C57"0=N&B=C87&>": C5vª2;C57">"v"2;7A, .103:UC57&2;D3S"2;03:A@37&B3:UC57&B=[6M303:UC57&0!N&B=C87&>":U6¦]YC5>&@3_~N&B3B=[6C?_;N&T:B3:6 0!N"C5M=0324!28Z=28B=<@ P{ξ ∈ B} "v;4!2;7c6B3:A©2;C87&>".: =5Z=2~628B37dN&0=B3[¦|SC5:"[\7&<3D Ω = {x , . . . , x } (C =8Z=2;62;B37dN&03B=[6<ª>2803:A@37&B=:C57~@=6< P(x , )=p suZ=vw=N"D3B=Nd@?>"28Z=<3w3<3B!Nª:M303254!25Z3@32;7"C}@?C8:":7&B3:"ª28B3<=2;6 ξ(x ) = x , .103ª0=N"C8M303254!25Z=2;B3<@^, +, , ( ,"! 4!Z3@?w=<=CZ!NvUC5M328|:"> µ > n B328_~N&><=C5<=6[¦|<=C5M3[\7dN"B3<@|>C}|U2;62 S®2;03B3v~Z3Z=< J 1 n i i P{µ = m} = Cnm pm (1 − p)n−m , J {1, 2, . . . , N } I3, , # ! # !-,..! # +# # # B=N P{ξ = m} = 1 , N m = 0, 1, . . . , n. m = 1, 2, . . . , N. '31'rpm&^µXr Y'#r# $p j#r *^ N&7&2~6N&7&<3w=2;C5T:2Y:A©q<4N&B=<32uCZ!vUw!N&D3B3:Dk>"28Z=<3w3<=B3[ M303:UC57&0!N&B=C87&>"2 (Ω, U, P) :M3032425Z3@32;7"C}@TN&T{C5v66N Mξ = i i X ξ = ξ(ω) B=N >"2;03:A@37&B3:UC57&B3:U6 ξ(ω)P(ω) .12;0328w3"25Z!<3w3<3B=N ξ M303<3B=<=6N&2;7 _8B!NAw328B3<3@ x , . . . , x C1>"2;03:A@37&B3:UC57~@=6"B=:C57"`CJ M (ξ + η) = M ξ + M η ,&.1:F<3B4vUT=3<3 :"D?TU:B=C57dN"B37&[ c PL2;CZ!< ξ ≥ η ]37&: M ξ ≥ M η ]>qw!N"MC57"(cξ) B3:C87&=<^]cM2;C5Z=ξ,< ξ M≥ c0 =< c.M ξ = 0 ]=7&: P{ξ = 0} = 1 , QS2;D=C57&><37&28Z=`"B3:!]p2~CZ=< ξ ≥ 0 < M ξ = 0 ]7&: ∀ω ∈ Ω 0 CZ!24!v287 ξ(ω) = 0 , V\6N&7&2~6N&7&<=w32;C8TU:2k:A©q<4N"B3<32 ξ >"[\0=N&©N"287"C}@?w328032;_k_;N"T:"B?0=N"C8M3032425Z=2;B3<@µCZ=vw=N"D3B3:D?>25Z=<=w3<3B3[ ξ 1:"0!6v;Z!:"D ω∈Ω ω∈A 1 k Mξ = K C;N"6:64!28Z=2"] ξ = P x I n i=1 i {ξ=xi } k X xi M I{ξ=xi } = @YB328T:"7":"0=N&@1vB3T33<@{< i=1 g(x) xi P{ξ = xi } = , 1_FC5>:"D=C87&>{+ ¯H9CZ!24!v287 i=1 Mξ = UL2;CZ!< k X η = g(ξ) M η = M g(ξ) = k X k X xi p i . i=1 xi P{ξ = xi }. C5Z=vw=N&D=B=Nd@>25Z=<=w3<3B=N=]37&: k X i=1 i=1 g(xi )P{ξ = xi }. k 1 ;+ P QS:TN"_8[\>UN&287C}@9TN"T?<{C5>:"D=C87&>":qV]=B=NA4!:7&:"Z=`"T:9>":UC5M3:"Z=`"_;:">N"7&`C}@q0=N">"2;B=C57&>:6 g(ξ) = .1:"Z!N&5N&@ g(x) = xn ]3M=:&Z=vw=N"2;6 k X g(xi )I{ξ=xi } . k X xni P{ξ = xi }. i=1 M ξn = .1vUC57&`?>{|:~4!2 =8TC8M32803&Z3@32~6:"8: N 0=N&_]^CZ=vw=N"D3B=Nd@O>"28Z=<3w3<3B!N ξ _8B!NAw328B3<=2 x M303<=B3<=6N&2;7 0!N&_"] . . . ] x − N 0=N&_,!s®0324B3282F_;B=NAw328B=<32"]M30=<3B3<=6N"2;6:"2FC5Z=vw=N&D3B=:"D?>"25Z!<3w3<3B3:D{<=6282;7><4 N i=1 1 1 k k x̄ = k k X Ni 1 X x i Ni = xi . N i=1 N i=1 OuC5Z=< N >25Z=<=TU:<*=8TC8M32803:~4!<37"C}@%>:~4!B3<|{<7&28|{©ª2SvUCZ=:>"<@|]!7&::7&B3:C8<37&28Z=`"B=N&@%w=N"C87&:"7dN "Z=<3_8TNªT{>"280=:A@37&B3:C87~@=6 p = P{ξ = x } ,3.1:Q;= 7&:6vC503254!B32821_;B=NAw328B=<32 x̄ T:"Z=28"Z=287"C}@?:"T:&Z!: M ξ i Ni N i x̄ = k k X 1 X x i Ni ≈ xi p i = M ξ N i=1 i=1 M N&T3<=6O:"0=N"_8:6¦]&6N"7&2;6N"7&<3w32~C5T:"2\:A©q<4)N&B3<32\C87AN"7&<=C87&<3w32~C5T3"[\>UN&287C}@ªTN&T9C503254!B32;2_8B=NAw=28B3<32\C5Z=vI@ w=N"D3B3:D>25Z=<3w=<3B3[F, B=N&_;[\>N"287"C}@ @ 6 #' )%-, .# $p jUpr U±p µN"7&2;6N"7&<3w32~C5T:"2G:A©q<4N"B3<32 % ξ , '& n #'& B=N"_8[\>UN&287C}@ MM|ξ|ξ , E :"_;B=NAw3<=6 nM ξ w32;0328_ m , #' !-, ( #'& n ) ! + ! , B=N&_;[\>N&2;7"C}@ M (ξ −m) < , ' #' ! , ( #'& n ) ! + ! , , M |ξ−m| 12;B37&0=N&Z=`"B=[\D6:62;B37ª>7&:"0=:"8:SM3:"0U@4!TN M (ξ − m) B=N"_8[\>UN&287C}@ + # ! # <:":"_;B=NAw=N&2;7"C}@9w328032;_ ,!¬1:"032;B3`ªT=>NA40=N&7&B=[\D<3_G4<=C5M32;0=C5<=< √Dξ B=N"_8[\>N"287"C}@ ! # +# ! ., + !-,' %-#! ' ! ## , Dξ QS ª<=6<%C5>:"D=C87&>N6< J +A = M ξ − (M ξ) I Dξ < 7&:}4N<{7&:&Z=`T:ª7&:}4N3]T:}4N = c} = 1 HUYDξ 4=Z3@?≥CZ > 0 :"D?DξTU:=B=C507dN"B37&[ c <=6"2 > 762~C57&:q0=N">"2;B=C57&>UP{ξ N n n n n 2 2 2 & & D(cξ) = c2 Dξ, D(ξ + c) = Dξ ; PLb1280=N">"28B!C57&>:¬1:SC8TU:8:!,AQkZ3@?ZC> [¦|q4!>"vU|>25Z=<=w3<3B ξ < |M ξη| 6 p !#1'Rh^z x >2803:A@37&B=:C57"`9C8v66[ C8:""[\7"<3D!, P( n [ Ak ) = k=1 n X k=1 P(Ak ) − X M ξ 2 M η2 . X P(Ak1 Ak2 ) + 16k1 28B=C87&>": I(Ak )− +AV X X I Ak 1 Ak 2 + IAk1 Ak1 Ak3 − · · · + (−1)n−1 IA1 A2 ...An . E C57dN&Z=:C8`>[\w3<=C5Z=<37&`?6N&7&2~6N&7&<3w=2;C5T:2k:A©q<4N&B=<32F:"28<|?w=N"C87&28D%0=N">"2;B=C57&>UN9<µ>":UC5M3:"Z=`"_;:">UN&7&`UC}@288:Nd434!B3:C87&`Qª> , 16k1 _8B=NAw=28B3<@^]!7&:C8<=C57&2~6NCZ!vUw!N&D3B3[¦| OuC5Z=< @®C5Z=vw=N&D3B!Nd@{>"25Z!<3w3<3B=N9< C5:"[\7&<=D A ξ= {ξ = x } ] i = 1, . . . , n ]#{x@3>&Z3, [email protected];.7",C}x@ }0!N&_8<328B=<32;6¦,.1:"0=:A©ª4)N&2;6v > CZ=vw=N"D3B3:"D >"28Z=<3w3<3B=:"D ξ NdZ!82803vq"v;4!2~6t:":_8B=NAw=N"7&`Sw=28032;_ α , B328_~N&>"<=C5<=6[°M3:03:A©4!28FB @ $p j#r ^ suZ=vw=N"D3B3[\2ª>"25Z!<3w3<3B3[ B3[\2F<=6<%N&Z=82;"03[ α , . . . , α ]37A, 2,4=Z3@?ZC>c"[¦| B ,ξ. .,..,.B. , ξ ⊂ R >[\M3:&Z!B@3287"C}@?0=N&>28B=C87&>": 1 i n i ξ 1 ξ1 m 1 ξm n n Y P{ξ1 ∈ B1 , . . . , ξn ∈ Bn } = k=1 QªNd4!<=6 =8T3><3>N&Z=28B37"B3:"2G:"M30=24!28Z=28B3<=2GB=28_;N">"<=C8<=6:C87&<^, $p j#r ¢^ .1vC87&` ξ M303<3B=<=6N&2~6[\2?_;B=NAw32;B3<@ B328_~N&>" [¦| x , . . . , x ] 1 6 j ξ ,...,ξ P{ξk ∈ Bk } i 1 m 1j1 njn P{ξ1 = x1j1 , . . . , ξn = xnjn } = n Y k k=1 ] {xi1 , . . . , ximi } 6 mk P{ξk = xkjk }. =1'h h ) %-, # .# % # .,.. )%-, # # % η = g (ξ ) ' ,"!. # # ,.ξ , . ./ . , ξ *! , ,' # ( ' QS2;D=C57">"<37&28Z=`"B=:=] / 1 i .1:Q=;7&:6v , i , suZ!vUw!N&D3B3[\2{>25Z=<3w=<3B3[ n gi (x) % # ) ! # '& i {ηi ∈ Bi } = {gi (ξi ) ∈ Bi } = {ξi ∈ gi−1 (Bi )}. Agi (ξi ) ⊂ Aξi lj r#$pj#r#^'r#(!!&()=n&()/1'3G'3rpm&^ / Xr Y'#rp QS:TN"_;N"7&25Z=`UC57&>:ªC5Z=24!v >cS28D{7&2;:"032~6[ :M3<30=N"287"C}@?B=NZ=2;66v 11'¨h ! ) '&( )%-, # .# % # ., % ) ! g (x , . . . , x ) g (y , . . . , y ) ξ , .) .%-., ,ξ, η# ,.#. . . , % η ζ = g (ξ ,/ . . . , ξ ) ζ = # ,. / g (η , . . . , η ) =1'Oh^-* ) %-, # .# % ξ , . . . , ξ # ,. $#'& 1 1 2 1 1 n 2 1 n 1 m 1 m 1 *! , ,' # ( ' .1vUC57&` ξ < / n n Y M ξ1 . . . ξn = M ξi B328_~N&>"B3:C87&<%6N&7&2~6N&7&<3w=2;C5T:8:9:A©q<4N&B=<@<=62;2;6 ξη = m k X X xi y j I A i B j , M ξη = xi yj P(Ai Bj ) = i=1 j=1 i=1 j=1 k X m X m k X X xi yj P(Ai )P(Bj ) = k X xi P(Ai ) m X yj P(Bj ) = M ξ M η E S<3DSC5Z=vw=N&Dª6:A©9B=:\4!:TN"_;N"7&` M3:c<3B4!vT3=<3<^]d2;C5Z=:C8M3:&Z=`_8:>N&7"`C}@ B32;_;N">"<=C8<=6:C87&`Q> ξ < η x C86¦,Z!2;66v?I3, +A, !#1'µh-± xzNd434!<37"<3>"B3:2GC8>":D=C57&>:14!<=C8M3280!C5<3<=C5<=6[F, F62;2;6 η D(ξ + η) = M ((ξ + η) − M (ξ + η))2 = M ((ξ − M ξ) + (η − M η))2 = Wcw3<37&[\>UNd@^]w37&: ξ − Mξ < M (ξ − M ξ)2 + M (η − M η)2 + 2M (ξ − M ξ)(η − M η). B328_~N&><=C5<=6[F]L4N9CZ=254!v"2;7 , E "ª<3D{CZ=vw=N"DM3:"Z=vw=N&2;7"C}@M=:<=B4!vT33<3<, ,..+ ! # ! / D(ξ η) # =! Dξ # !-+,.Dη .! #+ # ## B=N P{ξ = m} = Mξ = n X k2 = N +1 2 1 , N Dξ = n = 1, 2, . . . , N 7AN"TTN"T N2 − 1 , 12 1 1 1 (n + 1)3 − (n + 1)2 + (n + 1) . 3 2 6 ! # ! / , ( # !-,..! #+ # ## 4=Z3@{w3<=CZN9vC5M=25|:> k=1 .103254C57dN">"<=6 ηn >9><4!2FC5v66[ M ηn = {1, 2, . . . , N } µn > n B32;_;N">"<=C8<=6[¦|<=C8M3[\7dN&B3<@| P{µn = m} = Cnm pm (1 − p)n−m . ξ1 + · · · + ξ n n X ]3}4!2 ξk M ξk = np Dηn k=1 <3B4!<=TN"7&:"0?vC8M325|NS> k@ :6t<=C5M=[\7AN"B3<3<^, M :"}4)N X 7dN&T?TN&T = nDξ = npq, k k=1 Dξk = M ξk2 − (M ξ)2 = p − p2 = pq. ! # ! # ! )+# #' ! %-# ! # !-,..! #+ # ## , 1. vUC57&` ξ @Yw3<=C5Z=:/"25 Z![¦|ªN&0=:">>ª>[\":"0=TU2c:"NP^2;6N n <3_1v03B3[ C N ªN"0=N"6<^]<3_1T:"7":"03[¦| 3>287dN3, M :"}4N , P{ξ = m} = m n−m CM CN −M , n CN m 6 min{n, M } M N&03:>S"28Z=:";: +da .103254C57dN">"<=6 ξ c> ><4!2¦C5v66 ξ +· · ·+ξ , }4!2 ξ ¯1<3B4!<3TN&7&:0ªC8:""[\7"<@ J k # %-#&# % , ! , F6282~6 M ξ = ,=.1Q: =;7&:6v M ξ = P M ξ = n ,UQªNdZ=2;2"] ξ = P ξ + P ξ ξ ,11C8B3:=]w37": M ξ = M ξ , K¡M30=24!M3:"Z=:A©ª2;B3<3< k < l <=6282~-6 J 1 M N k n n k M N k k=1 n 2 k k=1 2 k k l k6=l M ξk ξl = M I{ξk =1,ξl =1} = P{ξk = 1, ξl = 1} = P{ξk = 1}P{ξl = 1|ξk = 1} = K¡032;_8v~Z=`d7dN&7"2"] X M ξ2 = M ξ + < M ξ k ξl = n k6=l M (M − 1) N (N − 1) M M (M − 1) + n(n − 1) N N (N − 1) M Dξ = M ξ − (M ξ) = n N 2 k 2 M 1− N N −n . N −1 iª!(1&{#'3"$p jUprµdjlm'np#o ()j#r#m#rp#&j#lm'n#poq(U#3=#o .1vUC57&` CZ=vw=N"D3B3[\2R>"25Z!<3w3<3B3[EB!N ] M303<3B3N"7&` #' ! ,.!$#+ # ## CZ=vw=N"D3B3[¦|F>"28Z=<3w3<=B ξ < η ,"s®:">U62;C57"B3[6%_;N"T:"B3:U60!N"C5M=0324!28Z=28B=<@ CZ!vUw!N&D3B3[¦|>"25Z!<3w3<3B ξ < η v;4!2;6 B=N&_;[\>N"7&`ª>2803:A@37"B3:C87&` 1 1 m ij P{(ξ, η) ∈ B} = i X xi ,yj ∈B n j P{ξ = xi , η = yj } = }4!2 B CZ >c":"21M3:~4)6B3:A©ª2~C57&>:ª> R , N"C8M303254!25Z=2;B3<32F>2803:A@37&B=:C57"28D?:"M303254!25Z=@3287"C}@?7AN"&Z=<=328D X pij , xi ,yj ∈B 2 x1 x2 ... y1 p11 p21 ... y2 p12 p22 ... ... ... ... ... 1_ª4!>v62;03B3:8:_~N&T:"B!N0=NC5M303254!25Z!28B3<@X6:A©qB3:M=:&Z=vw3<37"`?:~4B3:62;03B3[\D_~N&T:"BR0=N"C8M303254!25Z=2;B3<@4=Z3@ CZ!vUw!N&D3B3:D>25Z=<=w3<3B3[ ξ J P{ξ = x } = P p <4!Z3@{C5Z=vw=N&D=B3:"D?>"28Z=<3w3<3B=[ η J P{y = y } = P p , ¥ B=NdZ=:8<3w=B3[6/:"0=N&_;:64=Z3@ n CZ=vw=N"D3B3[¦|q>"28Z=<3w3<3B:"M303254!25Z=@3287"C}@ n@ 6280=B3[¦|q_;N&T:Bq0=N"C8M3032425Z=2;B3<@ i ij j j ij i Pξ1 ,...,ξn (B) = P{(ξ1 , . . . , ξn ) ∈ B}, }4!2 B@L6B3:A©ª2~C57&>:ª7&:w328T n@ 62803B3:8:M303:UC57&0=N"B=C57">N R , suZ=vw=N"D3B3[\2R>"25Z!<3w3<3B3[ ξ , . . . , ξ "v;4!2;6 7"0=N&T37":">N"7&`tTN"T T:6M3:B328B37"[ CZ=vw=N"D3B3:";:>"2;T37&:0=N , (ξ , . . . , ξ ) E M30=24!28Z=28B3<=2FB328_~N&>"25Z=<=w3<3B n 1 1 n ξ = n P{ξ1 ∈ B1 , . . . , ξn ∈ Bn } = n Y P{ξi ∈ Bi } :":"ªN&2;7"C}@B=NqC5Z=vw=N&D3B=[\2F>"28T=7&:"0=N ]3}4!2 , QkZ3@{7AN"T3<|?B328_~N&>""28T=7&:"03:>ªv;4!287qC5M=0=N&>24=Z=<=>Nª7&28ξ:03=2;6(ξN=, , . . . , ξ ) B ∈ R =1'h ) %-, # .#! '& !-, ξ , . . . , ξ # ,. g (x) % # ) ! # '& i )%-, # # % ηk = gk (ξk ) i1 # ,. / imi 1 i=1 1 n mi k +~ '(r&rcoqµ&j#lm'n#poq(jrpm#rp#o{% - - r#²#r#U#%^=##j²pr#r suZ!vUw!N&D3B=N&@ >25Z=<=w3<3B=N B=N"_8[\>N"287"C}@ B3:"0!6<303:">UN&B3B=:"D^]2~CZ=< < ,u.1vUC57&` ξ <=62;287 T:"B=28w3B3:2G6N"7&2;6N"7&<3w32~C5T:"2F:A©qξ<4)N&B3<321<4!<=C8M3280=C8F< >S,suZ=vw=N&D=B=Nd@>25Z=M<=w3ξ<3B==N 0 Dξ = 1 ξ − Mξ ξ1 = √ Dξ CGB3v~Z=2;>"[6 6N&7&2~6N&7&<3w=2;C5T=<=6 :A©9<4N&B3<=2;6<{254!<3B3<3w=B3:"D4!<=C8M3280!C5<32;DB=N"_8[\>UN&287C}@B3:0=6<303:>N&B=B3:"D^, .1vUC57&` < °C5Z=vw=N&D=B3[\2®>25Z=<=w3<3B3[ < < ¯1<3|SB3:"0!6<303:">T3<^,d¬1:Q=~F1<33<32;B37T:"0=0325Z3@3=<3< 628©ª4v ξ < η ξ:M303η254!25Z3@32;7"C}@?C8:":7&B3:"ª28B3<=2;6 ξ η 1 1 ρ(ξ, η) = M (ξ1 η1 ) = ρ(ξ, η) M (ξ − M ξ)(η − M η) √ . Dξ · Dη G1<=C5Z=<37&28Z=`9B3:C8<379B=N&_;>N"B3<32GTU:>N"03<=N&=<3<628©ª4v ξ < η <:":"_;B=NAw=N"287"C}@ Cov(ξ, η) = M (ξ − M ξ)(η − M η). 1_1B32;0=N&>28B=C87&>N9¬1:SL< @ SLvUB3@3TU:>C8TU:8: ξ η ) ©21C8>":D=C57&>:6:A©qB3:ª4!:TN&_;N"7&`B328M3:UC503254C57">"28B=(M B3:CJ 1 1 6 M ξ12 M η12 C5Z=24v"287A]3w=7&:q>C828}4N 0 6 D(ξ1 ± η1 ) = M (ξ1 ± η1 )2 = 2 ± 2ρ(ξ, η). |ρ(ξ, η)| 6 1 ,£®7&: , E 0=N&7&B=:"2?B328>2803B3:!,#.1vUC57&` ξ < η B=28_;N">"<=C8<=6[\2{CZ=vw=N"D3B3[\2 ρ(ξ, η) = 0 E , 3 M 3 5 2 ! 4 5 2 ! Z = < 6 , M :}4N ξ < ζ _;N">"<=C8<=6[F]>"::""ª2c;:">:"0U@^,b1: Mξ = Mη = 0 ζ = ξη OuC5Z=< ξ < η B=28_;N">"<=C8<=6[F]7&: >25Z=<3w=<3B3[ < 2 M ξζ = M ξ 2 η = M ξ 2 M η = 0 = M ξM η . ! # ! .1vUC57&`{CZ!vUw!N&D3B3[\2>"28Z=<3w3<=B3[ ξ < η 0=N">"B3:U62803B3:0=N"C8M303254!25Z=2;B3[B=N C8<=6[F,=.1:&Z!:A©9"<=C525Z=<3w=<3B3[ ξ < η B=N"_8[\>NQ> 7"C}@?B328T:030325Z!<303:">UN&B3B=[6<^, ¬1:N=;F1<3=<328B37ªT:"030328Z3@33<3"28B?24<3B3<33217&:}4N<7":&Z=`TU:97&:}4N3]T:"}4)NªC8vUª2;C87&>"v > 7w3<=C5Z!N < b 7AN"T3<32]w37&: P{η = aξ + b} = 1 , #"+ ' ( ,=.1vC87&` ] < , M :"}4)N P{η = aξ + b} = 1 M ξ = α Dξ = β a 6= 0 2 ρ{ξ, η} = M ξ − α aξ + b − aα − b β |a|β = sign(a). , M : }4N D(ξ − η ) = 2(1 − ρ(ξ, η)) = 0 , .1:N=87&:U6v , P{ξ − η = const} = 1 ¦.1vC57"` ρ(ξ, η) = −1 , M :}4N D(ξ + η ) = 0 < P{ξ + η = const} = 1 , r#n#' "# #U#(' 'rpm&^' #&r# .1vUC57&` < C5Z=vw=N&D=B3[\2>25Z=<3w=<3B3[ < >28ª2;C57">"28B=B=Nd@ 1vB3T3=<@ B=N , suZ=254!v > ª282OM3:"B@37"<32X< B3<3©ª2;C5Z=24vI>cªξN&@7&η28:032;6NªB3:UC}@37ª:"S<=D|UN"0=N&T=g7&280<B321_~N&>"N"2;6:DC}|2~6[F, 1 ' ,'&*% '&( / q.1vC87&` ρ(ξ, η) = 1 1 1 1 1 1 1 1 +~ B=N&_;[\>N"287"C}@B!N&<Z=vw3ª<=6 M303<3"Z=<©ª2;B3<32~6 > $p j#r ^ suZ=vw=N&D3B!Nd@>25Z=<=w3<3B=N 8C 6[C5Z=29B=N&NA40=N&7&:>=])2;C5Z=< M (η − g(ξ))g(ξ)M30=<3B3<=6N"287{B=N&"2A@ Z=<3w=<3B=N g(ξ) B=N&_;[\>N&2;7"C}@RC80324!B=28T3>UNA4!0=N"7&<3w32~C5T:"DR0328;032;C;C5<32;D/CZ=vw=N"D3B3:"DO>"28Z=<3w3<=B3[ η B!N?C5Z=vw=N&D=B3v >y>2R@ Z=<3w=<3B3v ξ , =1'h^-¢ # , ! # + # ! ., + ! ,.' %-# ! , ! # ) ! # g(ξ) # % η ,".# % ) ξ ##' 2 + + #( # # g(ξ) = m2 + ρ # # σ2 (ξ − m1 ). σ1 m1 = M ξ m2 = M η σ12 = Dξ σ22 = Dη ρ = ρ(ξ, η) *! , ,' # ( ' bFN"D4!2~6 _8B!NAw328B3<=2", F 6282~6 / < 7dN&T3<=2uw37&: α β ! #' ! ! ! # Φ(α, β) = M (η−α−βξ) 2 / M30=<3B3<=6N"287 B!N&<=62;B3`"ª282u>:"_~6:A©9B=:"2 Φ(α, β) = M η 2 + α2 + β 2 M ξ 2 − 2αM η − 2βM ξη + 2αβM ξ = FC8C5Z=24v"2;6 1vB3T332 B=ND=8TC87&032;6v6-J σ 2 + m22 + α2 + β 2 (σ12 + m21 ) − 2αm2 − 2β(cov(ξ, η) + m1 m2 ) + 2αβm1 . Φ(α, β) ∂Φ = −2(m2 − α − βm1 ) = 0. ∂α E 7"CA>L4NB=Nd|U:~4!<=6¦]w37&: K =;7&:9©ª21>"0=2;6#@ ∂Φ = 2[β(σ12 + m21 ) − (cov(ξ, η) + m1 m2 ) + αm1 ] = 0. ∂β β= cov(ξ, η) σ2 σ2 = ρ , α = m2 − βm1 = m2 − ρ m1 . σ12 σ1 σ1 ∂2Φ ∂α2 ∂2Φ ∂α∂β ∂2Φ ∂α∂β ∂2Φ ∂β 2 = 2 2m1 M N"T3<=6 :"0=N&_;:6¦]M303<{B=N"D4!2;B3B3[¦| α < β 1vB3T33<3@ 2m1 2(σ12 + m21 ) Φ(α, β) g(ξ) = m2 + ρ = 4σ12 > 0. M303<3B3&Z=@3287"C}@Z=<3B32;D3B3:"D{032;8032~C8C8<328D η B=N ξ , ¬1N: =;F1<33<=28B37 β = ρ B!N&_8[\>UN&2;7"C}@ ! #'& ! #) !$# η B!N ξ ,=.10U@=6N&@ σ2 σ1 B=N"_8[\>N"287"C}@ ! y − m2 x − m1 =ρ σ2 σ1 ! # ) ! # η B!N ξ ,3b12;7&03v;4!B3:qB!N&D37&<{B=N"<=628B=`"S2;21_8B=NAw32;B3<32 M (η − α − βξ)2 & min M (η − α − βξ)2 = σ22 (1 − ρ2 ). ®£ 7dNF>"28Z=<3w3<=B=NFB=N"_8[\>N"287"C}@ ' ,'& % +# ! # >"28Z=<3w3<=B3[ :7&B3:UC5<37&28Z=`"B=: ,"N&_;B3:C87&` B=N"_8[\>N"287"C}@O:C87AN"7&T:6 >"25Z!<3w3<3B3[ η :"7&B3:UC5<37"25Z=`B3:%>"28Z=<3w3<3B=[ ξ ,#ηsuZ=vw=N"D3B3[\2q>25Z=ξ<3w=<3B3[ η < ηξ =@3>&ηZ3@F−>cg(ξ) 7"C}@ B32;TU:030328Z=<303:>N&B=B3[6<^,3K C;N"6:6/4!25Z!2"] α,β 1 1 M (ξ − m1 )(η − m2 − ρ M (ξ − m1 )(η − m2 ) − ρ σ2 (ξ − m1 )) = σ1 σ2 M (ξ − m1 )2 = cov(ξ, η) − ρσ1 σ2 = 0. σ1 I"g ¥ B=NdZ=:8<3w=B3:q6:A©9B=:ª0=NC8C;6:"7&032;7&`9Z=<3B=28D3B3v > 032;8032~C8C8 >"28Z=<3w3<3B=[ ξ B=N>"25Z!<3w3<3B3v η J L(η) = m1 + ρ ¬1N: =;F1<33<=28B37k0=2880=2;C8C8<3< ξ B=N η 0=N&>28B ª28B3<32~6 ρ σσ12 σ1 (η − m2 ). σ2 ]<M30U@=6Nd@Z=<=B328D3B=:"D?0328;032;C;C5<3< ξ B=N η :"M30=24!28Z3@3287"C}@?C5:":7&B3:Q@ 1 x − m1 y − m2 = σ2 ρ σ1 E C87AN"7&:"w=B=Nd@q4<=C5M32;0=C5<3@?>25Z=<3w=<3B3[ ξ :"7"B3:C8<37&25Z!`"B3: η 0=N">"B=N σ (1 − ρ ) , E 2cM=0@=6[\2F0=2880=2;C8C8<3<{M303:A|U:~4=@37Sw32;0328_F7":"w3T3v (m , m ) <{C8:">M=NA4NQ> 7ª7&:&Z=`T:M=03< 2 1 1 !#1'Rh^ ^ ") 2 2 U'(U&()% =o/U(' ρ = ±1 , x>0 ##' #' &# ! , .# ' ., P{|ξ| > x} 6 M |ξ| x P{|ξ − M ξ| > x} 6 *! , ,' # ( '& K®:UC5M3:"Z=`"_;v"2~6C}@B3280=N">"2;B=C57&>:6 Dξ . x2 / |ξ| = |ξ|I{|ξ|>x} + |ξ|I{|ξ| |ξ|I{|ξ|>x} > xI{|ξ|>x} . KL[\w3<=C5Z3@@?6N"7&2;6N"7&<3w32~C5T:"2F:A©q<4)N&B3<32G:"79:"28<3|w=N"C87&28D3=87":"8:9B32;0=N&>28B=C87&>N=]M3:&Z=vw=N"2;6 M |ξ| > xM I{|ξ|>x} = xP{|ξ| > x}. QS03v;:"2B32;0=N&>28B=C87&>":SM3:&Z=vw=N"287"C}@q<3_cM3280=>":";:=]2;CZ!<28;:kM303"25Z!<3w3<3B32 η = (ξ − M ξ) ] M3:U6B@^]w37&: M η = Dξ , NC8C86:7&03<=6yM3:UCZ=254!:">UN&7&28Z=`"B=:C57"`OC5Z=vw=N&D3B=[¦| >"28Z=<3w3<=B ;, SLv;4!2~6yC8w3<37dN&7"`=]Yw37": :M3032425Z=2;B3[B=Nk>2803:A@37&B=:C57"B3:6OM303:UC57&0!N&B=C87&>"2 (Ω , U , P ) ]M30=<3ξw32;,6Xξ M=, .03.<. TNd©4!:U6 k < n CZ=vw=N"D3B3ξ[\2, ξ>25, Z=. . 7q0=NC5M30=24!28Z=28B3<=2F>"280=:A@37&B3:C87&28D]UB=2F_;N&><=C}@3S2;2c:7 n , , ! # ! @Y<=B4!<3TN&7":"0{vC8M325|N @ 8:<=C8M3[\7dN&B3<@{M303<{<=C8M3[\7dN&B3<3@|>C}|2~62 SL280=B3v~Z3Z=<^] bFN C8N"6:U6/ 4!25ξZ=2]~CZ > Iv >tM3:C5Z=24!:>N"7&25kZ=`B3:C87&`\B32;_;N&><=C525Z=<=w3<3Bk6:A©qB3: :M303k2564!25nZ=<=7&` B=N:~4!B=:62;C8TU:B328w=B3:6/>2803:A@37"B3:C87&B3:6tM303:UC57&0=N"B=C57">"2", suZ!24!v > Nd@?7&28:032;6N<=62;287q:"S<|q|N&0=N"T37&2;0^, !#1'Xh^ x G128[\S2;>U, ξ , ξ , . . . # ,. )& #' ) #' '&,"! , ! ' ,' , c > 0 # 2 1 n n 2 1 2 n k k % '& # Dξn 6 c n = 1, 2, . . . # ! lim P 1>P 2 o n ξ +...ξ M ξ1 + · · · + M ξ n 1 n < ε = 1. − n n *! , ,' # ( '& E " :"_;B=NAw3<=6Rw32;0328_ / n→∞ 1 ε>0 ζn = ξ 1 + · · · + ξ n <M303<=62;B3<=6/T ζn n o n ξ M ξn c Dξn n − 1− 2 2 >1− 2 , n n x n x n 7dN&T?TN&T Dξ = P Dξ 6 nc ,3K M3032425Z=2FM3:"Z=vw3<=6tB3vU©9B=:"2F0=N">"28B!C57&>:=, suZ!24!v > Nd@?7&28:032;6Nª@3>&Z=@3287"C}@{CZ=254C57&><32;6 M303254![L4!vS2;D^, n k n k=1 >"7":"03:2B3280!N&>"2;B=C57">": G12;"[\S2;>N =1'Xh^-³ x _~N&T:"B{:&Z=`S<|w=<=C528Z{>1:0=62!G128[\S2;>N5, ! "!-,.! # + # #$# M ξn = a # Dξn = σ2 < ∞ # '&&.! x > 0 lim P I3+ ξ1 , ξ2 , . . . # ., "+ , n ξ +···+ξ o 1 n −a < x = 1. n K =w N"C87&B3:C87&<^]4!Z3@?C}|U2;6[ SL2;03B3v~Z3Z=<{>"2;03B=N =1'Xh^z´ x SL280=B3v~Z3Z=0 ]5}42 ¯ <3B4!<3TN&7&:0 *! , ,' # ( ' .10=24C87AN">"<=6 >\>"<42YC5v66[XB328_~N&>""<=2", &j!(!µ1'3G'3rpm&^µXr Y'#rp N"C;C86:7&03<=6yCZ=vw=N"D3B3[\2%>25Z=<3w=<3B3[ < ]uM303<3B3 ª<32%_8B!NAw328B3<3@ < , WYCZ!:">"B=[6 _~N&T:"B=:60!N"C5M=0324!28Z=28B=<@ η M30=<%_;ξNA4)N&ηB3B3:U6t_;B=NAw32;B3<3< ξ = x B!N&_8:>"{x2~6,B=.N".".:", 0{x vUC}Z=:>"{yB3[¦|, .>.28.03, :Ay@3?7}@ B3:UC57&2;D n i i 1 n i n 1 m 1 n k P{η = yl |ξ = xk } = P{η = yl , ξ = xk } , l = 1, . . . , n. P{ξ = xk } ξ = xk WYCZ!:">"B=[6t6N&7"2;6N&7"<3w32;C8T3<=6 :A©q<4N"B3<32;6 η M30="28Z=<3w3<3B=[ M (η|ξ) ]@3>"Z3@F> ª28D=C}@1vB3T3=<328Dµ:7 ξ , suZ=vw=N"D3B3v >°>"25Z!<3w3<3B3v "v;4!2~6B=N"_8[\>N"7&`SvUCZ=:>"B3[6t6N&7&2~6N&7&<3w=2;C5T=<=6t:A©q<4)N&B3<32~6 η M=03<_~NA4N"B3B3:D CZ!vUw!N&D3B3:D>25Z=<=w3<3B32 ξ ,M (η|ξ) µN"7&2;6N"7&<3w32~C5T:"21:A©q<4N"B3<32FCZ!vUw!N&D3B3:D>25Z=<=w3<3B3[ M (η|ξ) 7&:}4N0=N&>B3: l=1 k M [M (η|ξ)] = F6282;7q62;C57":CZ=254!v > S2;21vU7">"280U©4!2;B3<32", =1'Oh^z! # ! !-,..# ' *! , ,' # ( ' F62;2;6 m X k=1 P{ξ = xk }M {η|ξ = xk }. M [M (η|ξ)] = M η. / M [M (η|ξ)] = m X k=1 m X n X yl P{η = yl , ξ = xk } = K¡032;_8v~Z=`d7dN&7"2FM303<|U:~4!<=6T%1:0=6v~Z=2 k=1 l=1 P{ξ = xk }M {η|ξ = xk } = Mη = m X n X l=1 yl P{η = yl } = M η. P{ξ = xk }M {η|ξ = xk }, w=NC57&B3[6 CZ!vUw!N&2;6tT:"7&:03:"D@3>&Z3@32;7"C}@?1:0=6v~Z!NM3:&Z!B3:"D?>"2;03:A@37&B3:UC57&<^, =^'j #'3 $## Yj #'3t3=1' l'(!' '$j'3' k=1 II !#1'Oh^z3h ) '&( ) # , p # 0 #,0# ##' #' !-,..# x' µ √ σ = npq → ∞ m−np √ m npq *! , ,' # ( '& .1vC57"` / P{µ = m} = √ m = np + xσ P{µ = m} = P{ , F6282~6 x2 1 1 e− 2 (1 + O( )) σ 2πσ n! µ − np = x} = pm q n−m , σ m!(n − m)! KL:C8M3:&Z!`"_8v2;6C}@1:0=6v~Z=:"Dµs®7&<30Z!<3B35N n! = √ }4!2 2πn nn e−n eθn , 11C8B3:=]w37": n ] m = np + x√npq = np(1 + ,3.1:N=87&:U6v n → +∞ xq σ ) < |θn | < q = 1 − p. 1 . 12n √ n − m = nq − x npq = nq(1 − xq σ ) C57"032;6<37C}@{T +∞ M30=< log P{µ = m} = log n! − log m! − log(n − m)! + m log p + (n − m) log q = n log n − m log m − (n − m) log(n − m) + m log p + (n − m) log q + M N&T{TN&T 1 n = log npq − log(1 + log m(n−m) xp σ ) log P{µ = m} = −m log − log(1 − xq σ ) 1 n log + θn − θm − θn−m . 2 2πm(n − m) = 2 log σ1 + O( σ1 ), B=N&|:~4!9Z!:"TNdZ=`B3:"D7&28:032;62F%vUN&>0=N @ kN&MZN"C8Nkv84:">&Z!287&>:"03<37"25Z=`B=Nd@^]2;C5Z=< m − np 1 = o((npq) 6 ) √ npq 287"C87&>"2;B3cB3>: 0 <{M3vU√C57&` |a| < 6 c < |b|√6 c , ]A}4!2 @7dN&T:2uB=N"<=628B3`S2;2 < 328Z=:"2]w37&: x m6]x[ ]3mN [x] 7dN&T:"2GB=N&<=":&Z!`"S2;2c=25Z=:2"]]np+a w37&: [x] 6npq[x , [np+b npq] ]x[ .1:"Z=:A©q<=6 m = np + x √npq ]7&:}4N ∆x = x − x = ,=s®:" Z!NC5B3:FZ!:"TNdZ=`B3:"DqM=0324!28Z=`"B=:"D 7&2;:"032~62 G12;0328_ 1 2 m m m+1 √1 npq m m2 m2 X X 1 2 1 µ − np 1 √ e− 2 xm ∆xm (1 + o( )). P{a 6 √ 6 b} = P{µ = m} = npq σ 2π m=m1 m=m1 ®s M30=N">N>0=N">"2;B=C57&>2FC57&:<37q<3B37&2;80=N&Z=`"B!Nd@%C5v66N3]C}|:~4=@3Nd@=C}@0=N">"B3:U62803B=:M=: a "2;03B3:90=N">"28B!C57&>: K¡:"S2~6tC5Z=vw=N&21M3vUC57&` 1 √ 2π ξn = µ−np √ npq Zc e− x2 2 Z∞ e− x2 2 dx = 1, −∞ 1 dx = 1 − √ 2π ,3Wcw3<37&[\>UNd@7dN"T©ª2]3w37&: −c Z e− x2 2 dx. |x|>c P{|ξn | > c} = 1 − P{|ξn | 6 c}. .1:Q=;7&:6v KL[\"2;032;6 σ→∞ 1 P{|ξn | > c} − √ 2π ,3bFN&D42;6 >0 Z e− x2 2 7dN"TU:2"]w37&: c>0 |x|>c 1 √ 2π 1 dx = P{|ξn | 6 c} − √ 2π Z e− x2 2 dx < Zc e− 8 |x| n1 <=62;2;6 1 P{|ξn | 6 c} − √ 2π Zc e− x2 2 dx < −c . 4 [A, B] = [a, b] ∩ [−c, c] KL[\"2;032;6y7&2;M32803`tZC> :"D <=B37&280=>NdZ < :":"_;B=NAw3<=6 B=N"D4!287C}@ n 7dN&T:"2]w37&:4=Z=@>C825| n >[a,nb] "28B!C57&>: 1_1B3280!N&>"2;B=C57">N 2 1 P{ξn ∈ [A, B]} − √ 2π 1 P{ξn ∈ [a, b]} − √ 2π n > n1 . 8 P{|ξn | > c} 6 2 Zb a 2 e dx . −c <{_;N1<3TC5<303v2;6/2;8:!,3.1:7":&Z=`TU:w37&:4:"TN&_~N&B3B3:U6vqB=N&D4!287"C}@ n ]w=7&:4=Z3@{>C825| .1:Q=;7&:6vM303<{7&25|©ª2 x2 2 − x2 ZB e− x2 2 dx < ,s®:" Z!NC5B3:O4:"TN&_~N&B3B3:U6v . 2 A 1 dx 6 P{|ξn | > c} + √ 2π Z |x|>c e− x2 2 dx I&P M3:"Z=vw=N&2~6¦]w37&:qM30=< 1 + P{ξn ∈ [A, B]} − √ 2π ZB e− x2 2 dx A n > n0 = max(n1 , n2 ) 0=N">"B3:U62803B=:ªM3:9>UC52;6 a 6 b. b12;0=N&>28B=C87&>": SL280303":"Z3@3287{:328B3<=7&`7&:w3B3:UC57&`µM=03<3&Z!<©ª2;B3<@>µ<3B=7&2R@ ;0=NdZ=`B3:"D7&28:032;62kµvN&>0=N @ kN&M3Z!N"C;N3, SL2;_%4!:TN"_;N"7&25Z=`UC57&>UN/C81:0=6v~Z=<303v2;6 N&B=N&Z=:"7&28:032;6[µvN&>0=N@ kN&MZ!NC8NR4=Z=@ M3:&Z!<3B3:6&Z=2;B3<3D ¯q8:ªC8:"[\7&<@> n <=C5M3[\7dN"B3<@|] i = 1, . . . , k ]C >"280=:A@37&B3:C87&`Q>¨vC8M325|N ]p µ + · · · +µµ = n ,!N"62;7&<=6¦]3w=7&:i4=Z3@ ∞ µ−np √ npq n 2 − u2 −∞ i i 1 k µi − npi xi = √ npi qi >[\M3:&Z=B3@3287"C}@C8:":7&B3:"ª28B3<=2 k X √ xi npi qi = 0 . !#1'Oh^z± , ( # # .# ! ' ' ( "+ # 0 < pi < 1 # '& ! ,"! , , # ! , ! +,.) ,.' ( G , ) # ! ! '& Pk xpii√np !-,. # ! i qi = 0 i−1 '&'&# ( G ! n→∞ ##' #' & ' %&## √ ),+ # dx q1 · . . . · qn P(G) → q Pk (2π)k−1 i=1 pi qi # #' .# , / Z exp G i=1 n o 1X qi x2i dx , 2 i=1 k − j!' 3U!U1' l!'3"!#' .1vUC57&` µ ¯q<3B3:U6<=NdZ=`B3:0=N"C8M3032425Z=2;B3B=Nd@{CZ=vw=N"D3B=N&@>"28Z=<3w3<=B=N3] m n−m P{µ = m} = cm , np q *) !#1'{h^z xz.1vUN"C;C5:"B!N, m = 0, 1, . . . ##' #'&& '& n% #→ ∞# p→0 q = 1 − p. ' ,"! #$% ' P{µ = m} = Cnm pm q n−m → np → a am −a e . m! #' ++ ") ! ! ., IV *! , ,' # ( ' QS2;D=C57">"<37&28Z=`"B=:=] / .103< n→∞ < P{µ = m} = np → a M3032425Z 1 2 m−1 (np)m (1 − )(1 − ) . . . (1 − )(1 − p)n−m m! n n n (1 − p)n 0=N&>28B e−a ,3.1:Q=;7&:6v P{µ = m} → am −a e . m! %:A©qB3:9M3:TN&_;N"7&`=]w37":q<=6282;7962~C57&:C5Z=24vI>cªN&@:"=28B3TN }4!2 a = np |P{µ = m} − , am −a a2 e | 6 np2 = , m! n # rGUUpr{$p j!#o t3!# .103<=&Z=<©28B3<=2"]=_~NA4N">N"2;6:"2F7"28:"0=2;6:"Dµ.1vUN"C;C5:"B!N3]B=N"_8[\>N"287"C}@ ) ,. . ! ,!.103<3"Z=<©28B3<32ªCkM3:Q@ 6:"ª`Q>°7&2;:"032~6 %vUN&>0=N @ kN&MZN"C8NkB=N&_;[\>N&2;7"C}@ ! , ( ,QkZ3@{0=N"C8M303254!25Z=2;B3<@{.1vN"C;C5:B=N P{µ = m} = <{<3B37&2;80=N&Z!N 1 Φ0 (x) = √ 2π am −a e m! Zx e− u2 2 B=N"_8[\>N"2;6:8:S<=B37&28;0=NdZ!:6 kN&MZ!NC8N=]U 7C}@?7dN&"Z=<33[F, .1vUC57&`!]B!N&M303"[\w3<=C5Z=<37&`?>"2;03:A@37&B3:UC57&` B32;_;N&><=C52803:A@37"B3:C87&`Q> vC8M325|N p , KL[\w3<=C5Z=<=6 x = ]x = {C}|2~62 SL2803B=v;Z=Z= n m √2 −np npq 1 P{m1 6 µ 6 m2 } ≈ √ 2π x m2 Z e− x2 2 dx = Φ0 (xm2 ) − Φ0 (xm1 ). 1. 03 6:A©qB3:1:"=28B3<37"`GM=:k1:"0=6v~Z=20SL28030=kM30=N">":Dªw=NC57&28B=C87&>N1B32~6B3:";: <3_~628B3<37"`qM3032425Z=[ <3B37&2;803<=03:">UN&B3<@^]M=:&Z!N"5Nd@ x m1 P{m1 6 µ 6 m2 } ≈ Φ0 (xm2 + 21 ) − Φ0 (xm1 − 21 ), } 4!2 x = < , .103254!25Z!`"B=N&@?7&2;:"032~6Nx %vUN&>"0!=NA¯ kN"MZ!NC8NFM3:_8>:&Z3@32;7S4)N&7&`M=03<3&Z!<©ª2;B3B3[\D%:"7">"287qB!NqCZ=254!v > S<=2k>":Q@ M303:UC5[1J .1vUC57&` µ ¯µw3M303< n <=C8M3[\7dN&B3<3@|>%C}|2;62DSL2803B=v;Z=Z=2803:A@37"B3:C87&`Q>yvC8M325|N p ] 0 < 3 ] >9TN&©4!:6t<=C5M3[\7dN"B3<3<^, p<1 , G12;6v90=N">"B=N1>2803:A@37&B=:C57"`k7&:";:=]w37&:w=N"C87&:"7dNFB=NC57&vMZ!28B3<@C8:"[\7&<@q:"7&T3Z=:"B@32;7"C}@:"7S>"280=:A@37&B3:C87&< B32F:&Z=2;2cw32~6tB!N α m2 + 12 m2 +1/2−np √ npq m1 − 21 m1 −1/2−np √ npq n r r o o n n µ n µn − np n n −p ≤α =P −α ≤ √ P ≈ ≤α n pq npq pq √n α pq Z rn x2 1 dx = 2Φ0 α ≈√ exp − . 2 pq 2π √ n −α pq I" ,=¬FN&T:2 B=N"<=628B3`S2;2Gw=<=CZ=:9<=C8M3[\7dN&B3<=D?B=v©qB3:9M30=:"<3_;>"2;C87&<4=Z3@{7&:";:=]w37&:"[ Cc>2803:A@37"B3:C87&`Q>ª]B32 62;B3`"ª28D β 3] w=NC57&:7ANB=NC57&vMZ!28B3<@%C5:"[\7&<3@:"7"TZ=:"B3@Z!N"C8`ª:7>2803:A@37"B3:C87&<B=2F":&Z!282Gw32;6 B=N α n µ rn o n P ≥β. − p ≤ α ≈ 2Φ0 α n pq | µnn ,3.103<4N"B3B3:"D>"2;03:A@37&B3:UC57&< β :"_~6:A©9B=[¦|9_;B=NAw32;B3<3D − p| P 1< 7D n µ rn o n =β. − p ≤ α ≈ 2Φ0 α n pq 8 % 4!% & 232 8 N/ 2 <5: 1< #$ # Y"jUpr{djlm#'3np=n (jrpm#rp#o .1vUC57&` (Ω, U, P) ¯M303:<3_8>:&Z=`B3:"2G>"2;03:A@37&B3:UC57&B=:"2GM303:C87&0=N"B=C57&>:=, ]®:M3032425Z=2;B3B=Nd@¡B=N <¡M303<3B3 Nd@ >28S2~C57&>28B3B=[\2 $p jUpr§ SvUB=T33<@ _;B=NAw328B=<@^]B=N&_;[\>N"287"C}@{CZ=vw=N"D3B3:"D?>"28Z=<3ξw3<==B3:"FD^]=(ω)2;CZ!<4!Z3@KZ > ":8: x ∈ RΩ 6B3:A©ª2~C57&>: {ω ∈ Ω : ξ(ω) ≤ x} = @3>&Z3@32;7"C}@?C5Z=vw=N&D3B=[6tC5:"[\7&<=2;6¦]37&:92~C57&`qM30=<3B=NA4!Z=25©q<37 U , {ξ ≤ x} $p jUpr ³ SvB3T33<@ F (x) = F (x) = P{ξ ≤ x} B!N&_8[\>UN&2;7"C}@1vB3T33<32;D0=NC5M30=24!28Z=28B3<3@ C5Z=vw=N&D3B=:"D?>"25Z!<3w3<3B3[ ξ , .1:qC8>":"D!C57&>vqB328M30=2803[\>B3:C87&<>2803:A@37"B3:C87&< P <=62;2;6 P(ξ < x) = F (x − 0) ,=.1Q: =;7&:6v P(x < ξ ≤ x ) = F (x ) − F (x ) h P(ξ = x) = F (x) − F (x − 0) * P(x ≤ ξ ≤ x ) = F (x ) − F (x − 0) ± P(x < ξ < x ) = F (x − 0) − F (x ) P(x ≤ ξ < x ) = F (x − 0) − F (x − 0) , !#1'µh ¢ ) ! F (x) # ) ., #' # F (x) #!$# ! , !-,.., ) + #'& ! #' '& ξ 1 2 1 & 2 & 1 2 2 & 1 1 2 2 1 & 1 2 2 1 % # F (+∞) = lim F (x) = 1, F (−∞) = lim F (x) = 0 . *! , ,' # ( '& 11C5B3:!]dw37&: Bn = {x < ξ ≤ x + 1 } ↓ ∅ ,".1:N=87&:U6v / , F 6282;6 P∞ ,=.1:N=8n7&:U6v An = {n − 1 < ξ ≤ n} Ω= An x→+∞ x→−∞ P(Bn ) = F (x + n1 ) − F (x) → 0 ,&K®>24!2~6 −∞ 1 = lim N →∞ N X P(An ) = lim N →∞ F (N ) − F (−N ) . ] $p jUpr h!´^ @ N&Z=82;"0=N M3:~46B3:A©2;C87&> ]^M=:"03:A©4!2;B3B=Nd@O<3B37&2;03>N&Z!N"6<4N =B N"_8[\>UN&287C}@":0325Z=2;>C8TU:D^,3£uZ=σ2;62;B37&[°":B0325Z=2;>C8TU:D σ@ NdZ=;R28"0!NSB=N"_8[\>UN?> 7C}@":0325Z!28>C8T3<=6<{6B3:A©ªa 2~UN"≤6<^b, O®CZ=< ¯C5Z=vw=N&D=B=Nd@q>"28Z=<3w3<=B=N3]&7":ªC5 @3>&Z=@3287"C}@ @ N&Z=82;"03:D^]&:":?@ _;B=NAw=N&2;7"C}@?C8<=ξ6>":"Z=:6 U <{B=N"_8[\>UN&287C}@ σ@ NdZ=;28"0=:"D^]M3:03:A©4!28{ξB=B3:"D%(B)CZ!vU, w!BN&D3∈B3:B}D{>"25Z!<3w3<3B3:D σ ξ , U @3>&Z=@3287"C}@ @ M3:~4N&Z=82;"03:D σ@ N&Z=82;"03[ U , σ !#1'Oh^z x 7&2;:"032~6Nq:qM303:~4:&Z3©28B3<3<%1vB3T3=<3<{>"280=:A@37&B3:C87& , x ]} M3:03:A©4N&2;7GN&Z=82;"03v B ]"C8:C57":A@3Sv ><3_\>C828>":_;6:A©qB3[¦|kT:"B=28w3B3[¦| C5v66°><4N P (x , x ] ,Ys®:"{(x :7&B3:S28B=<32 :M303254!25Z3@32;7 B=NªNdZ=;28"0=2 B σ@ NA4=4!<37&<3>B3v >¨1vB3T3= 6B=P:A©ª((x 2;C87&>,N=x]"v;])4!:=>&Z=P{ξ 2;7&>":∈0@F>c(xS,v x> ]}vUC=Z=:>"F<(x@=6 )1−¯ F4(x,U.1) :ª7&28:032;62c: M303:~4:&Z3©28B3<3"2;03:A@37&B3:UC57&<%1vB3T3=<@{6B=:A©ª2;C87&>N M30=:~4!:&Z3©9N&2;7"C}@B=Nq6<3B=<=6NdZ=`B3v > @ NdZ=;28"0=v"]3C5:>"M=N?@ 4QN > Sv > C σ@ NdZ!82803:"D{:"0328Z=28>UC5T3<|?6B3:A©ª2~C57&> B ,=PKL2803:A@37"B3:C87&` P (B) = P{ξ ∈ B} ] B σ∈ B ]=C8:">"M!NA4N"2879C 6280=:"D^]3M=:"03:A©4!2;B3B3:"D?1vB3T33<32;D{0=N"C8M3032425Z=2;B3<@ F "Z3@3287"C}@?<3B37&2;80=N&Z=:6 F28285?N @ s®7&<3Z=7&`"2~C8N n k=1 1 1k 2 ξ 2k 1 2 1 2 2 ◦ ◦ 1 ξ ξ ξ Pξ (B) = M 03:"D=TN B3:D ξ , Z dFξ (x) . @3>"Z3@3287"C}@{B3:>"[6 >"280=:A@37&B3:C87&B3[6 M303:UC57&0!N&B=C87&>":U6¦]=M3:03:A©4!28B3B=[6 C5Z=vw=N&D3B=:"D{>"28Z=<3w3C825| B ∈ B ]~B=N&_;[\>N&2;7"C}@F0=N"C8M303254!25Z=2;B3<32~6 $p j#r h^! kvB3T=3<@ >"2;03:A@37&B3:UC57&2;D?C5Z=vw=N&D=B3:"D?>"28Z=<3w3<3B=[ ξP, (B) .1vUC57&` <=62;287\B32u":&Z!282C5w32;7&B3:8:\w3<=C5Z!N\7&:w328TF0!N&_80=[\>N ] = x } = p N"287"C}@14!"25Z!<3w3<3B3:D^]A8x:>", :"x0U@3, .7c. 7A. N"TP{ξ ©ª2]dw37&: ξ <=62;287c4!<=C8T3032;7&B3:"2 0=NC5M303254!25Z!28B3<32, ξ 1 2 ! # ! / , ( # ! , !$#+ # ## P{ξ = k} = Cnk pk q n−k , k = 0, . . . , n , p ∈ (0, 1), q = 1 − p . ! # ! / ) ,. .! # !-,..! # +# # # P{ξ = k} = ak −a e , a > 0, k = 0, 1, . . . . k! ! # ! / # #' ! %-#! # !-,..! #+ # ## s®v66<303v2;6Nd@%1vB3T33<3@ P{ξ = k} = p (1 − p)k−1 , k = 1, 2, . . . , p ∈ (0, 1) . pξ (x) B=N&_8[\>UN&2;7"C}@MZ=:7&B3:UC57&`qCZ!vUw!N&D3B3:D>25Z=<=w3<3B3[ ξ ]32;C5Z=< P{x1 < ξ ≤ x2 } = Fξ (x2 ) − Fξ (x1 ) = 11C5B=:=]w37&: , >97&:w3TNd|B32;M3032;03[\>"B3:UC57&:"D=C87&>q1vUB=T33<3"28Z=<3w3<3B!N ]=<=62A>cªNd@{MZ=:7&B3:UC57&`!]B=N&_;[\>N&2;7"C}@N"C5:"ZC> 7"B3:ªB=28M3032;03[\>"B=:"D{CZ=vw=N"D3B3:"D?>"28Z=<3w3":0@37ª7dN&T©2"]3w3ξ7": ξ <=62;287N&UC5:"ZC> 7&B=:ªB32;M3032;03[\>"B3:210=N"C8M3032425Z=2;B3<32",!.103< =;7&:6 Pξ (B) = Z pξ (x)dx . ! # ! ! , ( # )+,) .! #"!-,..! #+ # ## , us Z!vUw!N&D3B=N&@/ >"28Z=<3w3<=B=N ξ <=6282;7{B3:"0=6N&Z=`"B=:"20=N"C8M3032425Z=2;B3<329CªM!N&0=N6287&0=N6< ]2;CZ!< ξ N&UC5:&ZK> 7&B3:9B32;M3032;03[\>"B=N<{2821MZ!:"7&B3:UC57&`q<4 σ>0 B pξ (x) = √ ] (x−a)2 1 e− 2σ2 . 2πσ ! # ! ,. # ! # !-,.! # + # # # , us Z!vUw!N&D3B=N&@S/ >"28Z=<3w3<=B=N ξ 0=N&>B3:62;03B3:c0=N"C8M3032425Z=2;B=N B!N :7&0328_;T2 <4 pξ (x) = ] (a, σ) −∞ < a < ∞ [a, b] ]~2~CZ=< ξ N"C8:&ZC>c7&B3: B32;M3032;03[\>"B=N   1 , x ∈ [a, b] b−a  0, x ∈ / [a, b] . ! # ! ! ,.!$#+ # ## , suZ!vUw!N&D3B=N&@/ >γ25Z=<3w=<3B=N ξ &Z3@32;7"C}@>9>"<4!2    0, x < 0 pξ (x) = λα xα−1 −λx  e , x≥0  Γ(α) GC M=N"0=N"62;7&0=N"6< α > 0 < λ > 0 , K w=NC57&B=:C57"<^])M303< M3:&Z=vw=N"2;6MZ!:"7&B3:UC57&`M3:"TN"_;N&7"25Z=`B3:";:q0=N"C8M3032425Z=2;B3<@^, M :}4NC5Z=vw=N&D=B=Nd@ >25Z=<3w=<3B=N ξ B!N&_8[\>UN&2;7"C}@M3α:T=N"_;1N"7&25Z=`B3:S0!N"C5M=0324!28Z=28B=B3:"D^, Z=2828;:">":D 62;03[ B3v~Z=`=]Yw=7&: ]Y7&: O®CZ=< C8vUª2;C87&>"v287O7AN"T:"2%CZ=vw=N"D3B3:2µC5:"[\7&<32 C5Z=vw=N&D3B!Nd@>25Z=<3w=<3B=N ξ B!N&_8[\>UN&2;7"C}@C5<=B38v~Z3@303B=:"D{<Z=<{B<=∈62">BS2;D%C5<3B3;v~Z3@303B3:2F0=N"C8M3032425Z=2;PB3<32"(B) , =1 K :""ª2;6 CZ=vw=N"2cKZ > N&@?1vB3T3=<@0=NC5M30=24!28Z=28B3<3@ F (x) M303254C57dN">"<=6Nª>q>"<42 ξ F (x) = a1 F1 (x) + a2 F2 (x) + a3 F3 (x) , }4!2 ] ] ¯q4!<=C8T3032;7&B=Nd@%1vB3T33<3@?0=NC5M30=24!28Z=28B3<3@^] F (x) ¯1vB3T33<@{0=NC5M303254!2R@ Z=2;B3<a@^]3≥ a +Nd@?a MZ=+:"7"aB3:=C87&1`=] FF (x) ¯C5<=B38v~Z3@303B!Nd@%1vUB=T33<@?0=N"C8M3032425Z=2;B3<@^, (x) B=N"_8[\>N"287"C}@?":0325Z=2;>C8TU:D?1vB3T=3<328D]=2;C5Z=c":8: B ∈ B $p jUpr hh^ SvB3T33<3@ M3:"Z=B3[\D?M303::""0!N&_ g (B) M303<3B=Nd4=Z=28©9g(u) <=7 B , !#1'yh^z ^ ξ )%-, , .# % , g(u) !$# #.! , ) ! #&'& τ = g(ξ) i 1 2 3 1 2 3 −1 # '& ) %-, # % / *! , ,' # ( '& .1vC57"` B ∈ B , M :"}4N g (B) = B ∈ B , M N"TqTN&T τ = ξ (B ) ∈ U ]&7&: τ ¯ªC5Z=vw=N&D=B=Nd@ >25Z=<3w=<3B=N3, / ! # ! / .1vC87&` g(x) = x ] F (x) ¯N&UC5:"ZC> 7&B=:qB328M30=2803[\>B=Nd@%1vB3T33<@{0=NC5M303254!25Z!28B3<@µCFM3Z=:"7 @ −1 B3:UC57&`N> < p (x) M3:"Z=vw=N&2~6 ξ ,=.103< τ =ξ 2 −1 −1 1 <3_kC8:":"7"B3:"ª28B3<@ 2 x≥0 1 ξ √ √ √ √ Fη (x) = P{ξ 2 ≤ x} = P{− x ≤ ξ ≤ x} = Fξ ( x) − Fξ (− x) √ √ 1 pη (x) = √ pξ ( x) + pξ (− x) . 2 x !1Up#oq?#'3"$p jUpr# .1vUC57&` ¯%CZ=vw=N"D3B3[\2>"25Z!<3w3<3B3[F,KL28T=7&:"0 C8:CZ!vUw!N&D3B3[6UN&7&ξ`9, C.Z!. .vU,w!ξN&D3B3[6t>"2;T37&:"0=:6¦, SvB3T33<3@?><4N ξ I" 1 n ξ 1 , . . . , ξn Fξ1 ,...,ξn (x1 , . . . , xn ) = P{ξ1 ≤ x1 , . . . , ξn ≤ xn } B=N"_8[\>N"287"C}@1vB3T3=<328D0=N"C8M3032425Z=2;B3<@C5Z=vw=N&D3B=:"8:ª>"2;T37&:"0!N3,3sL1:"0=6v~Z=<30=v"2;6B328T:"7":"03[\2GC5>":D=C57">NS1vB3TF@ 3<3<{0=NC5M30=24!28Z=28B3<3@ J xzC5: Z!N"C8:">UN&B3B3:UC57&`, 1 lim F (x , . . . , x ) = F (x , . . . , x ) , 2 lim F (x , . . . , x ) = 0 B=2FvU[\>N"287S<{B32;M3032;03[\>"B=N9C8M30=N&>UNSM=:qTN&©4!:6vN"038v628B=7&v", 3 F (x , . . . , x ) , 4 F (+∞, . . . , +∞) = 1 ◦ xn →+∞ ◦ xn →−∞ ◦ ξ1 ,...,ξn 1 n ξ1 ,...,ξn 1 n ξ1 ,...,ξn ◦ 1 ξ1 ,...,ξn−1 1 n−1 n ξ1 ,...,ξn 5◦ P{x1 < ξ1 ≤ x1 + h1 , . . . , xn < ξn ≤ xn + hn } = ∆h1 ,...,hn Fξ1 ,...,ξn (x1 , . . . , xn ) = , .1:C5Z=24B3282FC5::"7&B=:"S2;B3<32 4!:"TN"_8[\>N"287"C}@?6287&:~4!:U6t6N"7&2;6N"7&<3w32~C5T:"D{<3B4vUT=3<3<^, N"C;C86:7&03<=6 NdZ=;2803v B >C828>:"_;6:A©qB3[¦|qT:"B=28w3B3[¦|:"Q P24!<3B=28B3<3D?M30U@=6:"v;:&Z=`B3<3T:">ª>"<4N ,)bFN"<=628B=`"ªN&@ @ NdZ!8280=N3]C5:~4!2;0U©N&Nd@ ]B!N&_8[\>UN&2;7"C}@ @ N&Z=82;"03:D :"0328Z=28>UC5T3<3| Π {a < ξ ≤ b } 6B3:A©2;C57">=, σ@ NdZ=;280=N B @3>&Z=@3287"C}@σ _;N6[\TN"B3<32~6N&Z=82;"03[ BB :"7"B3:C8<37&25Z!`"B3σ:/:M3280=N"3<3D B∪ < ∩ ]u>_5@37&[¦| >{C5w32;7&B3:6 w3<=C5Z=2",^.1:N=87&:U6v σ@ N&Z=82;"0=N U = {ξ ∈ B, B ∈ B } = {U = ξ (B), B ∈ B } ])M3:03:A©4!28B=B=Nd@ CZ!vUw!N&D3B3[6{>28T37":"03:U6 ξ = (ξ , . . . , ξ ) C8:~4!280U©q<37"C}@F><=C}|U:~4!B3:D σ@ NdZ!828032 U >2803:A@37&B=:C57"B3:";:\M303:C87&0=N"B=C57&>UN ] , Ω, U, P U ⊂ U £uZ=2~628B37"[ N&Z=82;"03[ B M303254C57dN">&Z3F@ >c7"C}@ >O>"<42:Q P^254!<3B32;B3<@tM3:M=N&0=B3:B32;M3280=2;C52;T?N > ª<|UC}@M=0L@ @ P 6:"v;:&Z=`B3<3T:">/><4N ,¦.1vC87&` ]¦}4!2 ¯ M3:M=N&0=B3:B32;M328032~C52;TQN > ª<32;C}@ M30U@=6:"v;:&Z=`B3<3T3<^, E M3Π03254!25{a Z!<=6 <1vξB3T3≤3C< > b }6B=:A©ª2;C87&>BN=]!M==:&Z=:A©q<3B> P (B)B= P P{ξ ∈ B } x C86¦, 5 ,!QS:T?N @ _8[\>UN&2;7"C}@^]w37&:q7dN&T{:M3032425Z=2;B3B=Nd@%1vB3T33<3@{6B3:A©2;C87&>N P B32F_~N&>"&Z=2;B3<@ B ∈ B >9>"<42 :"N P^24<3B328B=<@qM3:"M=N"03B3:kB=28M32;032;C828T?N >cS<|C}@M30U@=6:vU;:&Z=`B3<3T:">1< σ@ NA4=4!<37&<3>B=NkB!NkNdZ=;28032 B ,3s M3:U6:"SN` > 7&2;:"032~6[ :9M303:~4!:"Z3©ª2;B3<3<>"280=:A@37&B3:C87&<1vB3T=3<@ P M303:~4:&Z3©9N&287C}@24!<=B=C57&>28B3B=[6:"0=N&_;:6O4:1vB3T3=<3< >"2;03:A@37&B3:UC57&<B=N σ@ NdZ=;28"0=2 B <9M3:"Z=B3:UC57&N` > :M303254!25Z3@32;7"C}@1vB3T=3<328D0=NC5M303254!25Z!28B3<@ F , M N"TqTN&T U ⊂ U ] <=62;287q62;C87&:q0=N&>28B=C87&>": P (B) = P{ξ ∈ B} , M 03:D3TN R , B , P @3>"Z3@3287C}@µB3:">[6¡>"2;03:A@37&B3:UC57&B3[6 M303:C87&0=N"B=C57&>:6¦,KL2803:A@37"B3:C87&B=Nd@1vB3T=3<@ = B & N ; _ [\>N"287"C}@0!N"C5M=0324!28Z=28B=<32;6 >"2;03:A@37&B3:UC57&2;D?C5Z=vw=N&D3B=:"8:>"2;T37&:0=N ξ = (ξ , . . . , ξ ) , P !+!$#'&# ! ,.!$#+ # ## _;Nd4N&2;7"C}@ T:"B32;w3B3[6<Z!< C5w=287&B3[6B=N&:"03:U6R7":"w32;T &Z=2;7&>":0F@ >cS<| P p = 1 ,KL2;03:A@37&B3x:UC57&=`(xC5Z=v,w=.N&.D=. B3, x:";: ) C5:"[\7&<3@ B ∈ B :"M30=24!28Z3@3287"C}@{C5::"7&B3:S2;B3<32;6 1 P (−1)n+θ1 +···+θn Fξ1 ,...,ξn (x1 + θ1 h1 , . . . , xn + θn hn ) θ1 ,...,θn =0 n i=1 n i i i n n n n ξ 1 ξ n −1 n n n n i=1 i i i k k k ξ ◦ k k ξ n n ξ n n n ξ ξ ξ ξ ξ 1 k n k k Pξ (B) = X n k 1 n k pk . ") # ! ' '&( !-,..! #+ # # B=N&_;[\>N"287"C}@7dN&TNd@ 1vB3T33<3@ ] ] w37": , .GZ=:7&B3:UC57&`{0=NC5M303254!25Z!28B3<@XB32;:"7&03<==N&7&28Z=`"B!N3] p (x)p ≥(x)0 N"B=N ,.GZ=:7&B3:UC57&`ª0=N"C8M3032425Z=2;B3<@ p "@3_;N&B=[ C5:":7&B3:Q@ xk ∈B R Pξ (B) = RB pξ (x)dx, B ∈ B n pξ (x)dx = 1 vC5Z=:"><32;6 S2;B3<32~6 n ξ ξ ξ ξ Rn pξ (x1 , . . . , xn ) = ∂ n Fξ (x1 , . . . , xn ) ∂x1 . . . ∂xn >97&:"w=TN&|B32;M303280=[\>"B3:UC57&"2;03:A@37&B3:UC57&2;D!^>28T37":"0=N ! +# ! / , ( ]2;C5Z=<:B3:9<=6282;7qMZ=:7&B3:C87&`9>"<4)N ξ p(x1 , . . . , xn ) = p (ξ1 , . . . , ξn ) |A| 1 n/2 exp{− 2 Q(x1 , . . . , xn )} , 2π B=N&_;[\>N"287"C}@ # H g }4!2 Q(x , . . . , x ) = P a x x ¯M3:&Z!:A©9<=7&25Z=`B3:q:"M=0324!28Z=28B=B=Nd@%B32;>"[\03:A©4!2;B3B=Nd@%1:"0!6N3] |A | ¯:"M30=24!2A@ Z=<=7&25Z=`6N"7&03<3=[ k a k , B=N"_8[\>UN&287C}@0=N">"B3:U62803B3[6 >q:"Q@ ! # ! N"C8M3032425Z=2;B3<32kCZ!vUw!N&D3B3:8:9>"2;T37&:0=N Z!NC57&< Ω ⊂ R ]32~CZ=/ <{2;8:9MZ=:7&B3:C87&`90=N"C8M303254!25Z=2;B3<@?0=N&>B=ξN = (ξ , . . . , ξ ) n 1 n ij i j i,j=1 ij 1 n p(x1 , . . . , xn ) = ( n |Ω |−1 , x ∈ Ω , 0, x ∈ / Ω. K =;7&:6 CZ!vUw!N&2c4=Z3@?ZC> :";:ª:"0328Z=28>UC5T:";:ª6B3:A©2;C57">N B ⊂ Rn <=6282~6 |B ∩ Ω | . |Ω | Pξ (B) = '31'rpm&^µXr Y'#r#Rdjlm#'3np=n (jrpm#rp#o .1vUC57&` 1vB3T33<3@0=N"C8M303254!25Z=2;B3<@RC5Z=vw=N&D=B3:"DR>"25Z!<3w3<3B3[ ξ B=N>"2;03:A@37&B3:UC57&B3:U6 < ≤ x} M30=:C57"0=N&B=C87&>"2 F Ω,(x)U,=PP{ξ ]0=NC5M303254!25Z!28B3<321>2803:A@37&B=:C57"28D{CZ=vw=N"D3B3:"D?>"28Z=<3w3<=B3[ ξ , P (B), B ∈ B $p jUpr h=* N&7&2~6N&7&<3w=2;C5T=<=6:A©q<4)N&B3<32~6C5Z=vw=N&D3B=:"D>"25Z!<3w3<3B3[ ξ B=N"_8[\>UN&287C}@Sw3<=C5Z=: M ξ ] 0=N">"B3:2 Z ξ ξ Mξ = ξ(ω)P(dω) , 2~CZ=v"287A, .1vUC57&` g :"0328Z=28>UC5TNd@ 1vUB=T33<@B=N R < R |g(x) |P (dx) < ∞ , M N&T¡TN&T¡1vB3T33<=< g(x) x ∈ R < ] ]!0=N">"B3:<3_;62;03<=6[ >M303:C87&0=N"B=C57&>UNd|µCS62;03:"D R, B, P < Ω, U, P ])7&:2;C87&` P {g 6 t} = g(ξ(ω)) ω ∈ Ω 3 ] & 7 9 : <=62;28762;C87&:90=N&>28B=C87&>": P{g(ξ) 6 t} Ω ξ R ξ %2;0=N Z Z g(ξ(ω))P(dω) = g(x)Pξ (dx) . M3:"03:A©4N"287"C}@?1vUB=T33<32;D?0!N"C5M=0324!28Z=28B=<@ F ]3M=:Q=87":6v Ω Pξ ξ R ξ M g(ξ) = Z g(ξ)(ω)P(dω) = Z g(x)Pξ (dx) = Z xPξ (dx) = Z g(x)Fξ (dx). .1:UCZ=254!B3<3D?<3B37&2;80=N&Z%B3:C8<37ªB!N&_8>UN&B3<=2G<=B37&28;0=NdZN F 2;"2;5N @ s®7&<Z=7&`2;C;N3,K w=NC57&B3:UC57&<] Ω R Mξ = Z ξ(ω)P(dω) = R Z xFξ (dx) QkZ=@4!<=C8T3032;7&B3:"D%C5Z=vw=N&D3B=:"D?>"25Z!<3w3<3B3[ ξ <:"0328Z=28>UC5T:"D?1vUB=T33<3< g <=6282~6 Ω Z g(x)dFξ (x) = X R R g(xk )[Fξ (xk ) − Fξ (xk − 0)] = QkZ=@?N&UC5:"ZC> 7&B=:B=28M3032;03[\>"B=:"D{CZ=vw=N"D3B3:"D?>"28Z=<3w3<=B3[ ξ <=6282~6 R }4!2 k Z R k g(x)pξ (x)dx , ¯MZ!:"7&B3:UC57&`90=NC5M30=24!28Z=28B3<3@{C5Z=vw=N&D3B=:"D?>"25Z!<3w3<3B3[ ξ , R pξ g(x)dFξ (x) = Z X g(xk )P{ξ = xk } . =H + 1_1:"M=0324!28Z=28B=<@?<=B37&28;0=NdZN F2;"2;5NSCZ!24!v287A]3w37&:q6N"7&2;6N"7&<3w32~C5T:"2F:A©q<4N"B3<32 T:"B=28w3B3:7&:}4N< 7&:"Z=`"T:97&:"}4)N3]T:"}4N M |ξ | < +∞ x M ξ 6:A©2879C5vS2~C57&>:">UN&7&`!]UB=:B=2F"[\7&`9T:"B32;w3B3[6YM5, ξ M N&T3<=6 :""0!N&_8:U6¦]=2~CZ=< ¯4!<=C8T303287"B=Nd@C5Z=vw=N&D3B!Nd@µ>"28Z=<3w3<=B=N3]!7&: M ξ = P x P{ξ = x } ,HOuC5Z=< F <=62;2879MZ=:7&B3:C87&`=]=7&: M ξ = Rξ xp (x)dx , '& , ,' # ,.' %-# ! ") +,. J O®CZ=< P{ξ = c} = 1 ]7&: M ξ = c h^ M (ξ + ξ ) = M ξ + M ξ ]32;C5Z=<%C5vS2~C57&>Iv >c7k4>N<3_F7&0328|?6N"7&2;6N"7&<3w32~C5T3<|?:A©q<4N&B=<3D * O®CZ=< ξ ≤ ξ ]37&: M ξ ≤ M ξ ]3>9w=NC57&B=:C57"<^] |M ξ | ≤ M |ξ | ±p O®CZ="280=:A@37&B3:C87&Q` >y+ *! , ,' # ( ' 11C8B3:=]w=7&: {ξ > 0} = P{ξ ≥ } < {ξ ≥ } ⊂ {ξ ≥ } ,3.1Q: =;7&:6v / k k k ξ ξ R & 1 2 1 1 & 2 2 1 2 2 & 1 A & A & 1 n 1 n n 1 n+1 0 = M ξ = M ξI{ξ=0} + M ξI{ξ>0} = M ξI{ξ>0} ≥ M ξI{ξ≥ n1 } ≥ 1_FC5>:"D=C87&>NªB328M=032803[\>B3:C87&<{>"2;03:A@37&B3:UC57&"2;B=C57">": Dξ = M ξ − (M ξ) , ! # ! / 1. vC87&` ξ ¯XB3:"0!6NdZ=`B3:µ0=NC5M303254!25Z!28B3B=N&@ CZ!vUw!N&D3B=N&@/>"25Z!<3w3<3B=NOCM!N&0=N6287&0=N6< M :}4N P{ξ > 0} = 0 2 Mξ = < Z R 2 n (t − a)2 o 1 dt + a exp − (t − a) √ 2σ 2 2πσ Z exp R n − 2 (a, σ) (t − a)2 o dt = a , 2σ 2 OuC5Z=< ξ B=:"0=6N&Z=`"B3:0=NC5M303254!25Z!28B=NCGM=N&0=N6287&0!N"6< Dξ = σ . ]7": η = σξ + a B3:0=6NdZ!`"B3:90=NC5M303254!25Z!28B=NC (0, 1) M=N"0=N"62;7&0=N"6< (a, σ) , ! # ! / .1vC87&` ξ ¯CZ!vUw!N&D3B=N&@>25Z=<3w=<3B=N3]0!N"C5M=0324!28Z=28B=B=Nd@?M3:q_;N"TU:B3v.1vNC8C8:"B=NªC1M=N&0=N6287&0=:6 ,µ M :"}4N Mξ = < X µn n e−µ = µ n! n ! # ! / .1vC87&`C5Z=vw=N&D=B=Nd@?>"25Z!<3w3<3B=N ξ 0=N&>B3:62;03B3:90=NC5M303254!25Z!28B=NªB=N9:"7"0328_;TU2 Dξ = µ . 1 Mξ = a−b Zb xdx = a+b 2 < Dξ = [a, b] , M :"}4)N (b − a)2 . 12 ^ O®CZ=< 0 ≤ ξ % ξ ]=7&: M ξ → M ξ xz7&28:032;6N9:q6:"B=:"7&:B3B3:"D{C}|:~4<=6:C87&c":";: ω ∈ Ω >"[\M=:&Z=B@32;7"C}@{B3280=N">"2;B=C57&>: lim ξ = ξ x 7"28:"0=2;6N F28285Nª:q6Nd©ª:03<303v2;6:DC}|:~4<=6:C87&25Z=<=w3<3B=N ξ <=6282;7M3Z=:"7&B=:C57"` Z 7&: ] pξ (x) g(x) ¯":0325Z=2;>C8TN&@1vB3T33<@?< |g(x) |pξ (x)dx < +∞ , R M g(ξ) = Z g(x) pξ (x)dx . R '(r#"r#Goqµ&j#lm'n#poqµ()j#r#m#rp#o E 4B3<=6¡<3_ª>N&©9B=28D3ª<|%M3:"B@37"<3D7&2;:"03<=<µ>"2;03:A@37&B3:UC57&2;D%@3>&Z3@32;7"C}@µM3:B@37&<32B328_~N&><=C5<=6:UC57&<µC5Z=vw=N&DF@ B3[¦|>"25Z!<3w3<3B^, $p jUpr h=± suZ=vw=N&D=B3[\21>"25Z!<3w3<3B3[ ξ , . . . , ξ B!N&_8[\>UN?>c7"C}@ # ., ]U2;C5Z=< 1 n ]_;Nd4N&B3B=[\2B=N>"280=:A@37 @ p$ jUpr h ^ suZ=vw=N&D3B=[\2>"28Z=<3w3<=B3[ M3:C5Z=24:">N"7&25Z!`"B3:UC57&< B3:UC57&B=:6M=03:C87&0=N&B!C57&>2 Ω, U, P ]3B=N&_;[\>NQ> 7"C}@B=28_;N">"<=C8<=6[6<^]2~CZ=<4=Z3@?ZC{ξ>c"}:";:ªw=<=CZ!N k Fξ1 ,...,ξn (x1 , . . . , xn ) = Fξ1 (x1 ) . . . Fξn (xn ) . ∞ n n=1 E 76287&<=6¨7dN&T©2"]w37":4!Z3@RM3:UCZ=254!:">UN&7&28Z=`"B=:C57"<µB328_~N&><=C5<=6[¦|C5Z=vw=N&D=B3[¦|µ>"28Z=<3w3<3BO"v;4!v7?B328_~N&> [\2?4!>NXC8:""[\7"<@^]Y:"7"B3:C}@3ª<32;C}@ T¡B328M32;032;C828TN?c> S<=6C}@ 803vM3M!N"6yCZ=vw=N"D3B3[¦| >25Z=<=w3<3B 8= 7":"D Fξ1 ,...,ξk (x1 , . . . , xk ) = Fξ1 (x1 ) · · · Fξk (xk ) . M3:UCZ=254!:">UN&7&28Z=`"B=:C57"<^, !#1'h^z³^ ) ' ( # ,. # )%-, # .# % / F (x) #'&( ) ! !-,..! #+ # #&, '& #!$# ! ) %-, &.# % ξ ' '&( p (x) # i = 1, . . . , n # '& F # # ' & ' & ' ( # ! , ) p (x ) · · · p (x ) !-,'& #$# p (x , . . . , x ) (x , . . . , x ) F , & ' # $ ! # ! , * 2 # & ' & ' # , '& #!$# ! / ++ # i / F (x) ! '& (x , . . . , x ) ξ1 ,...,ξn 1 n ξ1 ,...,ξk 1 k . ξ1 , . . . , ξn ξi i ξ1 ,...,ξk 1 ξi k ξ1 1 ξn n ξi pξ1 ,...,ξn (x1 , . . . , xn ) = pξ1 (x1 ) · · · pξn (xn ) % '& . + ) / *! , ,' # ( '& uO CZ=25Z=<3w=<3B3[ N"C8:&ZC>c7&B3:GB328M30=2803[\>B=N / < Fξi (xi ) = 7&:qC8:">62~C57&B!Nd@?1vB3T=3<@0!N"C5M=0324!28Z=28B=<@?<4 Zxi pξi (ti )dti , −∞ Fξ1 ,...,ξn (x1 , . . . , xn ) = Fξ1 (x1 ) · · · Fξn (xn ) = Zxn Zx1 pξn (tn )dtn = pξ1 (t1 )dt1 . . . = −∞ −∞ = E 0=N&7&B=:=]M303254!M3:&Z!:A©928B=C87&>Nª<=6282;762;C87&:9M3:"w37"UCR>L4!v]M3:Q=;7&:6v?1vB3T33<3CA>L4!v", QªNd4!<=64!03v8:2c:M3032425Z=2;B3<32FB32;_;N&><=C5"28Z=<3w3<3B, B=N&_;[\>N"287"C}@FB32;_;N&><=C5c"[¦| $p j#r h!¢^ suZ=vw=N&D3B=[\2u>"28Z=<3w3<=B3[ ":0325Z!28>C8T3<| B ∈ B ] i = 1, . . . , n ]3<=62;287q62;C87&`q0=N&>ξ28B=, .C8.7&.>",: ξ −∞ −∞ ξi i 1 n i 1. 03<=>"2428B3B3[\2F:M303254!25Z=2;B3<@{B328_~N&><=C5<=6:UC57&<{CZ!vUw!N&D3B3[¦|?>"28Z=<3w3<3B&=;T3>"<3>UNdZ=2;B37&B3[F,QkZ3@?7":"8:!]w37&:"[ 4!:TN"_;N"7&` =;T3>"<=>NdZ!28B37&B=:C57"`ª>:C5M=:&Z=`_8v"2~6C}@CZ=254!v > ª28D{7&28:032;6:D7"28:"0=<3<%62803[F, $p j#rh ^ ¥\Z=;28"0=[ C5:"[\7&<3D < B=N"_8[\>U?N > 7C}@B328_~N&>"c"[¦| C5:"[\7&<=D A ∈ A < A ∈ A <=6282;762;C87&:q0=N&>28B=AC87&>": A P{ξ1 ∈ B1 , . . . ξn ∈ Bn } = P{ξ1 ∈ B1 }· . . . · P{ξn ∈ Bn } . 1 1 1 2 , )+# ! # .,.σ. =1'h^ h!´^ '& 2 2 U1 P(A1 A2 ) = P(A1 )P(A2 ) . U2 # ! + # #* ' .#' ' .# # .,.. , )+# ! , M 28M3A2;03`G6A[6:A©2;6{C81:0=6/ v~Z=<303:>N"7&`\v7&>"2;0©4!2;B3<32]54!:TN&_8[\>UN?>cSv > =8T3><3>N&Z=28B=7&B3:C87&`\:"M=0324!28Z=28B=<3D 1 2 I"Vª<%I&=, =1' h^ h^! )%-, # .# % ξ1 ξ2 # ,. &' "),+, ' (! '&*),+ ,*# ! "),+, # , # ! + # # '& )%-, .# % , Uξ ξ / < 3 B ; 2 ; _ " N " > = < 8 C = < 6 [ < < ¯ 3 M & : ! Z U v = < 3 B & 7 8 2 = > d N ! Z [ " > < 4 N < , F 6282;7 ξ1 ξ2 ∆ Λ (a1 , a2 ] (b1 , b2 ] σ , )+# ! U *! , ,' # ( ' 1. vUC57&` / 1 2 62;C87&:q32;M3:"w3TNª0=N&>28B=C87&> P{ξ1 ∈ ∆, ξ2 ∈ Λ} = P{ξ1 ∈ (a1 , a2 ], ξ2 ∈ (b1 , b2 ]} = QªNdZ!282"]2~CZ=< ] = Fξ1 ,ξ2 (a2 , b2 ) − Fξ1 ,ξ2 (a1 , b2 ) − Fξ1 ,ξ2 (a2 , b1 ) + Fξ1 ,ξ2 (a1 , b1 ) = = Fξ1 (a2 ) − Fξ1 (a1 ) Fξ2 (b2 ) − Fξ1 (b1 ) = P{ξ ∈ ∆}P{ξ2 ∈ Λ} . ∆i i = 1, . . . , n ]=< ] Λj j = 1, . . . , m ¯94!>21C8<=C57"2;6[°B328M=28032~C528TNQ> S<3|UC}@<3B=7&2803>UNdZ=:>=]7&: P{ξ1 ∈ ∪ni=1 ∆i , ξ2 ∈ ∪m j=1 Λj } = X X = P{ξ1 ∈ ∆i , ξ2 ∈ Λj } = P{ξ1 ∈ ∆i }P{ξ2 ∈ Λj } = i,j i,j M N"T3<=6 :"0=N&_;:6¦]3N&Z=82;"03[F]M3:03:A©4!28B=B3[\21M3:&Z=v<3B=7&2803>UNdZ!N6< 4!v2879B328_~N&><=C5<=6:UC57&` U "2;0©4!2;B3<32FCZ!24!v2879<3_10=N&>28B=C87&> ξ1 ∆i < = P{ξ1 ∈ ∪ni=1 ∆i }P{ξ2 ∈ ∪ni=1 Λj } . Λj B=28_;N">"<=C8<=6[F, 1_17&2;:"032~6[ I3, I"gªC5Z=2R@ ξ2 Fξ1 ,ξ2 (x1 , x2 ) = P{ξ1 ≤ x1 , ξ2 ≤ x2 } = =P{ξ1 ≤ x1 }P{ξ2 ≤ x2 } = Fξ1 (x1 )Fξ2 (x2 ) . £®7dN97&2;:"032~6NC8M30=N">"24!Z=<3>Nk4=Z3@?ZC>c":";:ªw=<=CZ!NqC5Z=vw=N&D=B3[¦|>"28Z=<3w3<=B^, H"P ikj "(!r ) '&( !$# #.! # ) ! # ,. #" )%-, #".# % / ' "),+ , η = ϕ (ξ ) η =ϕ ϕ, ϕ(ξ ) ' ,"!. ## ., #ξ ) %-, ξ #.# % / < .1:N=87&:U6v<=_cB32;_;N">"<=C8<=6:C87&< σ@ NdZ=;280 U <4!B3:!]w37": C5Z=24!v287B328_~N&><=C5<=6/ :UC57&` U ""28T=7&:"0=N ϕ (x , . . . 1 1 1 1 2 2 2 1 η1 η1 2 2 η2 ξ1 ξ1 ξ2 ξ2 η2 ] ª:"0328Z=28>UC5T3<321vB3T=3<3<^]#7&:C5Z=vw=N&D3B=[\2>"28Z=<3w3<3B=[ ϕ (ξ , . . . , ξ ) ] ϕ (η , . . . , η ) B32;_;N">"<=C8<=6[F, ,B3 28_~#N&%->,<=C5 <=# 6/ [F1]#_{7&:7&284=:Z303@t2;6[ :"03I28,JZ=I28+>UC5C5T3Z=<324|Xv"6287AB3]Y:Aw=©7&2;:XC87&2;> CZ!< T:6M=:"B32;B37&[ ξ < ξ C5Z=vw=N&D3B=:"8:O>"2;T37&:0=N ξ =] (ξ , ξ ) B = B × B = {(x , x ) : x ∈ B , x ∈ B } ]>[\M3:&Z=B3@3287"C}@?0=N&>28B=C87&>": B ,B ∈B . . . , xn ) ϕ2 (y1 , . . . , yn ) 1 n 1 m 1 1 1 1 2 1 1 2 1 n 2 1 1 n 2 1 2 1 1 2 2 2 P{ξ ∈ B} = P{ξ1 ∈ B1 , ξ2 ∈ B2 } = P{ξ1 ∈ B1 }P{ξ2 ∈ B2 } . :>":"0U@37ª7&:"}4N=]w37&:62;0=N 0=N&>B=NSM=03:"<3_;>"254!28B3°62;0 Pξ Z P ξ1 D=87":6 CZ=vw=N"216:A©287"[\7&`9_~N&M3<=C;N&B?>9>"<4!21M3:>"7&:03B3:8:<3B37&2;80=N&Z!NFJ B OuC5Z=< ]37&:}4N g(x , x ) = g (x )g (x ) ZZ g(x1 , x2 )Pξ1 (dx1 )Pξ2 (dx2 ) . B1 ×B2 1 2 1 Z 1 2 2 g(x1 , x2 )Pξ1 ,ξ2 (dx1 , dx2 ) = Z g1 (x1 )Pξ1 (dx1 )· M30="<3<]Uw=7&:q<3B37&2;80=N&Z=[°C}|:~4=@37"C}@^, !#1'h^ hh x 1:"0=6v~Z!N{T:6M=:"_8<=3<3U62;C57"B3:";:F0=N"C8M3032425Z=2;B3<@CZ=vw=N"D3B3:8:1>"2;T37&:"0!NFC\T:6M3:B328B37dN?@ 6< ξ < η 0!N&>"B!NªM=03/ :"<3_;>"254!28B3 pM3Z=:"(x, 7&B=:y)C57"28D p (x) "B=:6280=B3:90=N"C8M3032425Z=2;B3[ B=N:7&0328_;T2 −∞ ξ2 ( pξ1 (x) = pξ2 (x) = < pξ1 +ξ2 (z) = Z1 [0, 1] , M :"}4)N 1, x ∈ [0, 1] , 0, x ∈ / [0, 1]  / [0, 2] ,   0, x ∈ pξ1 (x − t)dt = x, x ∈ [0, 1] ,   2 − x, x ∈ [1, 2] . lj r#$pj#r#^'r#(!!&()=n&() =1'h h=* )%-, #".# % # '& Mξ Mη ! # +,.# η # ,. # ' ! # % # ,'&# ,'& %-# ξ 1. 03<=>"242;6 , ! *! +*! ,.,.'&# ( ' ., / suZ=vw=N&D=B3v > >"28Z=<3w3<3B=v ξ v842;6OB=N&_;[\>N"7&`FM303:UC57&:D^]"2~CZ="<=6N>9>"<42 M ξη = M ξ · M η . ξ= m X c k I Ak , }4!2 A ¯M3:M=N&03B=:B=2;C5:>62~C57&B3[\2FC5Z=vw=N&D3B=[\21C8:"[\7&<@^, +", OuC5Z=< ξ ≥ 0 ];7":GC8vUª2;C87&>"v2877dN&TN&@1M=:CZ!24!:>N&7"25Z=`B3:C87&`M303:UC57&[¦|FB32;:"7&0=<3=N&7"25Z=`B3[¦|SCZ=vw=N"D3B3[¦|F>"28Z=<3w3<=B ] ]3w37": ξ % ξ M3:w37&<{B=N">"280=B3:"2,K TNAw32;C87&>"2 ξ 6:A©qB3:90=N"C;C86:"7"03287&` ξ ξ ≥0 k=1 k n n n n n ξn = n2 X k−1 2n I{ k−1 n <ξ≤ 2 k 2n } . 11C5B=:=]w37&: <{M303< <=6282;7962~C57&:qB=280=N">"28B!C57&>: ≤ ξ (ω) + , I,).1vC87&` ξ 0< ≤η ξB328≤_~N&>ξ<=C5<=6≤[\ξ2SM303:C87&[\ξ29(ω) CZ=v≤w=N"nD3B3[\2S>25Z=<=w3<3B3[F])M303<=B3<=6N?>cSξ(ω) <32_8B!NAw328B3<3@ x < · · · < x < M , " : } ) 4 N y < ··· < y k=1 n n+1 n n 1 2n 1 1 n ξ= < m X k=1 xk I{ξ=xk } , η = n X k=1 m yk I{η=yk } K =;7&:6 CZ=vw=N"2"]TN&T?B=N6t<3_;>"2~C57&B3:!] M ξη = M ξ · M η x C86¦,37"28:"0=2;6v?I3, HU, H3,)bFN&M=:6B3<=6¦])w37":2;C5Z=< ξ < η B328_~N&><=C5<=6[ < g < g ¯{":"0=25Z=2;>C5T=<32ª1vB3T3=<3<^]7&:?C5Z=vw=N&D=B3[\2S>25Z=<3w=<3B3[ < 7dN&T©2FB328_~N&><=C5<=6[F, g (ξ) g (η) K :"":_8B=NAw=28B3<@|qM3vB3T=7AN{+1<=62;2;6 P{ξ = xk , η = yl } = P{ξ = kk } · P{η = yl } . 1 1 2 2 M (ξη) = lim M (ξn ηn ) = lim M ξn · lim M ηn = M ξ · M η . P=,K:""ª2;6°CZ!vUw!N&2 ξ = ξ − ξ < η = η − η ]#}4!2 B32;:"7&03<==N&7&28Z=`"B=[\21C5Z=vw=N&D=B3[\2F>"28Z=<3w3<3B=[F,3.1:Q=87":6v n→∞ + − n→∞ + − ξ± < B328_~N&><=C5<=6[\2TN"Tt1vB3T33<=< ξ < η < n→∞ η± M (ξη) = M (ξ + − ξ − )(η + − η − ) = M (ξ + − ξ − ) · M (η + − η − ) = M ξ · M η . H ikj "(!r ) %-, # .# % %-# ! # + , #,# '& ξ1 , . . . , ξn # .,.. # ' ! #% # ,' # ,.' M (ξ1 · · · · · ξn ) = M ξ1 · · · · · M ξn . dj!(!!G'33U1'r#mU"^!Or Y'pr .1vUC57&` (Ω, U, P) ¯>2803:A@37&B=:C57"B3:"2FM30=:C57"0=N&B=C87&>":!] B ⊂ U < P(B) > 0 ,N"C;C86:"7"03<=6 (Ω, U, P ) ]}4!2 ,!.1vUC57&` ξ C5Z=vw=N&D=B=Nd@?>"25Z!<3w3<3B=NB=N (Ω, U, P) ]N_8B=NAw3<=7ª<{B=N (Ω, U, P ) , P (A) = P(A|B) $p jUpr h ^ WYCZ=:>"B3[6O6N&7"2;6N&7"<3w32;C8T3<=6X:A©q<4N"B3<32~6CZ!vUw!N&D3B3:D9>"25Z!<3w3<3B3[ ξ :"7&B3:UC5<37"25Z=`Q@ B3:qC5Z=vw=N&D3B=:"8:9C5:"[\7&<3@ B ]=B=N&_;[\>N&2;7"C}@w3B3:"2 B B B MB ξ = M (ξ|B) = F62;2;6 Z ξ(ω)PB (dω) . Ω M (ξ|B) = Z ξ(ω)P(dω|B) = 1 P(B) Z ξ(ω)P(dω ∩ B) = Ω Ω 1 = P(B) }4!2 Z ξ(ω)P(dω) = M (ξ; B) , P(B) B M (ξ; B) = .1vUC57&` ¯0=N&_;"<32;B3<32 Ω , M :}4N {B } Z ξ(ω)P(dω) . B n Mξ = XZ n ξ(ω)P(dω) = X M (ξ; Bn ) = n X P(Bn )M (ξ|Bn ) . x I3, +A n £®7dNk1:0=6v~Z!NF@3>"Z3@3287"C}@q:":"S2;B3<32;6X1:0=6v~Z=[¡M3:&Z!B3:"Dq>2803:A@37&B=:C57"<94=Z3@6N"7&2;6N"7&<3w32~C5T:";:k:A©q<4N&B=<@^,UK C;N"6:6/4!25Z!2"]32~CZ=< ξ = I ¯<3B4!<3TN"7&:"0{C5:"[\7&<@ A ]37": M ξ = M I = P(A) < Bn A A M (ξ|Bn ) = M (IA |Bn ) = 1 P(Bn ) Z IA (ω)P(dω) = Bn 1 P(Bn ) Z P(dω) = ABn P(A Bn ) = P(A|Bn ) . P(Bn ) ^ N"628B=<3>9>{x I,-+~ ξ B=N I < M (ξ|B ) B=N P(A|B ) ]=M3:&Z=vw3"2;03:A@37&B3:UC57&<, ! # ! N&:"w3<3D¡:C5Z=vU©9<=>N&2;7 :~4!B3:7&<3M3B3[¦| C87AN"B3T:">!]u0=NC5M3:"Z=:A©ª2;B3B3[¦| M=0@=6:"Z=<3B32;D3B3: B=N 0=NC8C87&:A@3B3<3UN&2879<|%M3:~4!|:~4=@{TµB3<=6 >M3:"0U@4!T2k:w3280=24!B3:UC57&<,bFN"D37&< C80324!B=<3D{M3280328|:~4%625©4!vC87dN&B3TN"6<, G12;0328_ B :"":_8B=NAw=<=6C5Z=vw=N&D=B3:"2OC5:"[\7&<32{¯0=N&:"w3<=D B=Nd|U:~4!<37"C}@¡v¡C57dN"B3TNCµB3:U62803:U6 k , M :"}4)N ,UKL2803:A@37&B=:C57"`S7&:8:=]Uw37&::"7ªC57dN&B3TN 0=N&:"w3<=DB=N&M=0=N&>"Z3@3287"C}@qT?C57dN&B3T=v i ] i = 1, . . . , n ]7dN&T3©ª2 P(B ) = 1/n 0=N">"B=N 1/n ,UQkZ=<3B=NM32;0325|U:~4NSM=03&< =87&:U6 C8:C57dN">&Z3@32;k7 |i − k| < A n n k k M (ξ|Bk ) = .1:91:"0=6v~Z=2x I,-+~ k−1 X i=1 n X 1 k(k − 1) (n − k)(n − k + 1) 1 (k − i) + (i − k) = + n n 2n 2n Mξ = i=k+1 n X k=1 P(Bk )M (ξ|Bk ) = n X 1 k(k − 1) . n2 k=1 Ha M N"T{TN"T n X 7&: k=1 k(k − 1) = Mξ = 1 (n − 1)n(n + 1) , 3 1 n−1 1+ . 3 n M N"T3<=6y:"0=N&_;:6¦]2;C5Z=<:"UCZ=vU©q<3>N"287"C}@ n C87AN"B3T:">!]#7":C503254!B@@ 4=Z=<3B=NOM32;0325|U:~4NR0=N&:"w32;8:XC5:C87dN&>&Z=@3287 3M 03<=&Z=<3_;<37&28Z=`"B3: , .12;0328D42;6/T0!N"C8C;6:"7&0=28B3°N&UC5:"ZC> 7&B=:SB32;M303280=[\>"B3:8:k>28T37&:0=N ,LG1280=28_ "v;4!2;6:"":_8B=NAw!N&7&` <3B37"2803>UNdZ]&C8:~4!280U©N"S<3D y ,dWYCZ=:>"B=N&@ª1vB3T33<@0=N"C8M303254!25Z=2;B3<@9CZ=vw=N"D3B3(ξ,:"Dη)>"28Z=<3w3<=B3[ ∆ξ M303<vCZ!:">"<=< η = y :"M=0324!28Z3@3287C}@TN&T?M303254!25Z n−1 3 y OuC5Z=<%CZ=vw=N"D3B3[\D{>"2;T37&:"0 lim P{ξ 6 x|η ∈ ∆y } = Fξ (x|η = y) . "B!Nd@M3Z=:"7&B=:C57"`0!N"C5M=0324!28Z=28B=<@ 1 |∆y | 1 |∆y | pξη (u, v) R Rx ∆y −∞ R R∞ pξη (u, v)dudv = pξη (u, v)dudv Rx pξη (u, y)du −∞ R∞ . pξη (u, y)du −∞ ∆y −∞ pξ (x|η = y) pξ (x|η = y) = ]37&: B=Nd|U:~4!<37"C}@4!<=F1280=28B33<=03:">UN&B3<32~6 x I3, I pξη (x, y) d Fξ (x|η = y) = R∞ . dy pξη (u, y)du WYCZ!:">"B=:"2F6N&7&2~6N&7&<=w32;C8TU:21:A©9<4N&B3<=2 ξ M303<{_~NA4N"B3B3:6_8B=NAw32;B3<3< −∞ M (ξ|η = y) = Z∞ η=y >"[\w3"28Z=<3w3<3B=[ C5:"[\7&<3@| ω ∈ {η = y} , M N&T%TN&T −∞ pη (y) = <3_x I,JI"®B=Nd|U:~4!<=6¦]w37": Z∞ M (ξ|η) ]UM303<3B=<=6N&2~6:"2B=N!=5Z=2~628B37dN&0=B3[¦| pξη (u, y)du −∞ E 7"CA>L4NB3287&0=v84B3:qM3:&Z!vUw=<37&`=]=w37&:qM303c(x,"[¦y)| a=< pb (y)p (x|η = y) . ξη η P{a 6 ξ 6 b} = Z∞ pη (y) ξ Zb pξ (x|η = y)dx dy . F¬ N"Tq<94=Z3@C5Z=vw=N&D3B=[¦|>25Z=<3w=<3B^]U:"M303254!25Z!28B3B3[¦|9B=NFT:B328w3B=:6O>"2;03:A@37&B3:UC57&B3:U6OM303:C87&0=N"B=C57&>2"]<=6282;7 62;C87&:q0=N">"28B!C57&>: M M (ξ|η) = M ξ ,UQS28D=C87&>"<37"25Z=`B3:=] −∞ M M (ξ|η) = Z∞ Z∞ −∞ −∞ xpξ (x|η = y)dx pη (y)dy = a Z∞ Z∞ −∞ −∞ xpξη (x, y)dydx = Z∞ −∞ xpξ (x)dx = M ξ . H 1. :N=87&:U6vq<=6282;762;C57":q0=N&>28B=C87&>": Z∞ Mξ = x I, H pη (y)M (ξ|η = y)dy , T:"7":"03:2cTN&T?UN&7&`UC}@TN&T?:":""ª28B3<=2G1:0=6v~Z=[ M3:"Z=B3:D>2803:A@37&B=:C57"<^, −∞ !$ 3,.%-$32&:(,.2&%0: 32&%D8 232 7B $# YjU#r#{r¡$#!"3n Xr#%"(!n&()' '#'3p3U#r#"rpm&#r - l#p²#r#n .1vUC57&` ]}42 < 4!28D!C57&><37&25Z!`"B3[\2µC5Z=vw=N&D3B=[\2?>25Z=<=w3<3B3[F]v TU:7&:"0=:"D C8vS2;C87&>"v >c7O< T:"B=28w3B3[ M ξ ζ< M= ηξ,!+iηN&7&2~6N&7&<3ξw=2;C5Tη:2F:A©9<4N&B3<=2GT:U6MZ=28TC8B3:"D%CZ!vUw!N&D3B3:D{>"25Z!<3w3<3B3[ ζ :"M=0324!28Z=<=6C5::"7 @ B3:S2;B3<32;6 E C8B3:">B3[\2GC8>":D=C57&>UNª6N"7&2;6N"7&<3w32~C5T:";:9:A©9<4MN&B3ζ <3=@^]M>"TξCZ +>cw=iMNd@%ηC5.>:"D=C87&>NNA4=4!<37&<3>B3:C87&<{<%6v;Z!`d7&<3MZ=<=TN"7&<3Q> @ B3:UC57& 7C}@>C5<3Z=2",UQS:TN&©ª2~6/B3280!N&>"2;B=C57">": s®vS2~C57&>v"2;7k4!2;D=C57">"<37&28Z=`"B=:"2 α 7dN&T:"2]w37&: .1vUC57&` eiα ζ = ξ1 + iη1 1vB3T3=<@ ]3}4!2 ξ < 1 η1 |M ζ | ≤ M |ζ | . |M ζ | = M eiα ζ . ¯94!2;D=C57&><37&28Z=`"B3[\2kC5Z=vw=N&D=B3[\21>"25Z!<3w3<3B3[F, F6282;6 |M ζ | = M ξ1 ≤ M |ξ1 | ≤ M |eiα ζ | = M |ζ | . $p jUprh!³^ GN&0=N"T37&280=<=C57&<=w32;C8TU:D 1vB3T3=<328D CZ=vw=N"D3B3:"D >"28Z=<3w3<3B=[ 4!2;D=C57&><37&28Z=`"B3:8:M3280=2;628B=B3:";: t ]30=N">"B=N&@ f (t) ξ B=N"_8[\>N"287"C}@ ξ fξ (t) = M eitξ = M cos(ξt) + iM sin(ξt) = O®CZ=<%CZ!vUw!N&D3B=N&@>25Z=<3w=<3B=N ξ 4<=C5T30=287&B=N=]37&: fξ (t) = X Z∞ eitx dFξ (x) . −∞ eitxk P{ξ = xk } . O®CZ=<%CZ!vUw!N&D3B=N&@>25Z=<3w=<3B=N9N&UC5:&ZK> 7&B3:9B32;M3032;03[\>"B=N=]7&: k fξ (t) = Z∞ eitx pξ (x)dx , 7&:2;C87&`|N&0=N"T37&280=<=C57&<=w32;C8TN&@1vB3T33<@ f (t) >B=87&:U6 C Z!vUw!N&2ªC8:">M=NA4N"287CSM30328:"0=N"_8:>N&B=<32;6 SvU0=`"2SM3Z=:"7 @ B3:UC57&"28Z=<3w3<3B=[ ξ , #! ' ! # . ' ., , !-,"! ' # ! ' %-# ! ) ! |f (t) | ≤ 1 4!Z3@?>UC528| t ∈ R < f (0) = 1 , h^ f (t) 0=N">"B3:U62803B=:ªB32;M3032;03[\>"B=NM3: t , .10=24!>UN&03<37"25Z=`B3:k4:"TNd©2;6/Z!2;66v", −∞ ξ ξ ξ ξ ξ 11'Oh^ h^ ! + # ' '&# ( x # eix − n−1 X H n≥1 ##' #'& !-,..#' |x |n (ix)k ≤ k! n! *! , ,' # ( ' QS:TN&_;N"7&25Z!`C57">":SM=:<=B4!vT33<3<, F6282~6 / |e ix −1| = .103254!M3:&Z!:A©9"2;B=C57">":/>"2;03B3:M=03< KL:C8M3:&Z=`_8v2;6C}@C5::"7&B=:"S2;B3<32~6 Zx .1:Q=;7&:6v e ix − n X (ix)k eiu − = k! Zx ,QS:"TN&©ª2;628;:tC5M=0=N&>24=Z=<=>":C87&`/M303< m = n k=0 , m = n+1 k=0 e iu − n−1 X M 2;M3280=`qB3287&0=v84B3:4!:"TN"_;N&7"`C8>":"D!C57&>:9I3,=.1:&Z=:A©q<=6 k=0 Z|x | e du ≤ du ≤ |x | . iu n (iu)k 1 ix X (ix)k du = e − . k! i k! n−1 X Zx k=0 (iu)k du ≤ k! Z|x | |x |n+1 un du = . n! (n + 1)! , M :}4N A = {|ξ | ≤ x} k=0 |f (t + h) − f (t) | = |M eitξ eihξ − 1 | ≤ M |eihξ − 1|IA + M |eihξ − 1|IĀ ≤ ≤2M IĀ + |h |M |ξ |IA ≤ x|h | + 2P{|ξ | > x} . S<3TC5<30=v"2;6 <{>"[\28032~6 7dN&T^]w=7&:"[ ,=.1:"Z=:A©q<=6/7&2;M3280=` , M :}4NS4=Z3@{>C528| h ] 4=Z3@{T:"7&:03[¦| ε >|h |0< δ >[\M3:&Z=B3@3x287"C}@?B3280=N">"2;B=P{|ξ C57&>: ||f>(tx}+ "<37"25Z=`B3[\21TU:B=C57dN"B37&[F]37&: f (t) = e f (at) , < $p j#r*´^ ¬1:U6MZ=2;TC8B3:"_;B=NAw3B3[\2RCZ=vw=N"D3B3[\2O>"25Z!<3w3<3B3[ B32;_;N&><=C5" ξ [¦=|"ξ:"0=+25Z=2;iξ>C5T=<| ηB=, Bη +⊂ iηR >"[\M=:&Z=B@32;7"C}@0!N&>"2;B=C57">": ε 4 ε 2x η itb ξ 1 ξ1 ,ξ2 2 1 η1 ,η2 1 2 2 2 P{(ξ1 , ξ2 ) ∈ B1 , (η1 , η2 ) ∈ B2 } = P{(ξ1 , ξ2 ) ∈ B1 } P{(η1 , η2 ) ∈ B2 } QªN&B=B3:"2\:M3032425Z=2;B3<32B32;_;N">"<=C8<=6:C87&"25Z!<3w3<3B9Z=2;8T:k:":"ªN"287"C}@SB!NkCZ=vw=N"DqT:"B32;w3B3:";:1w3<=CZN T:6MZ!28TC5B3:_8B=NAw=B3[¦|CZ=vw=N"D3B3[¦|>25Z=<=w3<3B^, ±p O®CZ=< ξ , . . . , ξ B=28_;N">"<=C8<=6[F]7&: 1 n fξ1 +···+ξn (t) = n Y fξk (t) . SQ :TN"_8[\>UN&287C}@C1M3:6:S`Q>°6v~Z=`d7&<3M3Z=<3TN&7&<=>"B3:8:qC5>:"D=C87&>N6N&7&2~6N&7&<=w32;C8TU:8:9:A©q<4N"B3<@^, ^ E :"_;B=NAw3<=6¡w=28032;_ m = M ξ , OuCZ!< m TU:B328w=B3:=])7&:µC8vUª2;C87&>"v > 7{>UC529M303:<3_8>:~4!B3[\2 f (t) M3:0U@4!TN < k≤n x I3, P f (0) = i M ξ . ¬103:U62G7":"8:!] UN&>11:"0=6N&Z=`"B3:1|N&0=N"T37&2;03<=C57"<3w32;C8T3v > 1 vB3T33 / w3N"7&`_;N&T:B3B3:C87&`ª4!<=F12;0328B3=<303:">UN&B3<3@M3:~4µ_8B!N&T:6 <3B37&2;80=N&Z!N6:A©qB3:!]30=N"C;C5vU©4Nd@{M3:q<3B4!vT33<3<^, .1vUC57&`9C5::"7&B3:S2;B3<32qx I3, UuM=0=N&>:"Dµw=NC57&< x I3,Ja"M3:7"28:"0=2;62 F28"2;5Nq:6N&©ª:03<303v2;6:"DRC}|:~4!<=6:UC57&x I3, U t = 0 ] M=03<|:~4<=6 T x I, PU5, N&_5Z!:A©ª28B=<32¤x I,JV" CZ=254!v"2;7<3_?7&2;:"032~6[ M 28DZ=:0=N3, ¢ F628287962;C87&:q0=N">"28B!C57&>: f (−t) = f (t) , ]=7": :"M=0324!28Z=28B!NqM303< ,.10=< =87&:U6 |f (λ)| ≤ 1 ] f (λ) ])032;8v~Z3@303B!N> Imλ > 0 < ^ OuC5Z=< B32;M3032;03[\ξ>"B=≥N=]0>"TZK> fw=N&(λ) @80!N&B3<3=v Imλ = 0 ] Imλ ≥ 0 k itξ ξ e ξ eihξ − 1 h k+1 ≤ |ξ |k+1 ξ ξ ξ fξ (λ) = Z∞ ξ eiλx dFξ (x) . 1b 2;M3032;03[\>"B3:UC57&`ª4!:TN&_8[\>UN&2;7"C}@TN&T{>C5>:"D=C87&>"2kI3,3288v~Z3@30=B3:C87&`CZ!24!v2879<3_k>:"_;6:A©qB3:UC57&<4!<=F12A@ 032;B33<30=:">N"B3<@M=:~4%_;B=N&T:U6/<3B37&2;80=N&Z!N3, QSC5Z=vw=N&D=B3v > >"28Z=<3w3<=B3v ]!M30=<3B3<=6NQ> Sv >7&:&Z=`T:325Z![\2SB32;:"7&03<==N&7&28Z=`"B=[\2S_8B!NAw328B3<3@^] v842;6%B=N"_8[\>UN&7&` # *% # CZ!vUw!N&D3B3:Dkξ>"28Z=<3w3<=B3:"D^,&N"T:"BF0=NC5M303254!25Z!28B3<@ª325Z=:w3<=C5Z=28B3B=:"DSC5Z=vw=N&D3B=:"D >25Z=<3w=<3B3[ :M303254!25Z3@32;7"C}@>2803:A@37"B3:C87;@=6< pn = P{ξ = n}, n = 0, 1, . . . , 4=Z=@T:"7&:03:"D , p =1 $p jUpr*! # ! '&'& .! "+, &# P∞ n n=0 ϕξ (s) = ∞ X ) ! # B!N&_8[\>UN&2;7"C}@C5v66N90U@4N x I3, U pn sn , 0=N">"B=N&@ M s ,3@4µ>%x I3, ULN&C8:&ZC>c7&B3:qC}|:~4<37"C}@M30=< |s | ≤ 1 , OuCZ=< ϕ (s) = M s ¯qM=03:"<3_;>":~4=@3Nd@1vB3T=3<@=25Z=:w3<=CZ!28B3B3:D?C5Z=vw=N&D=B3:"D?>"28Z=<3w3<3B=[F]37&: n=0 ξ ξ ξ fξ (t) = ϕξ eit . 4!2;C8`ª©21vUTN&©ª2;6t28ª2G:~4B3:qC5>":D=C57">":M303:"<=_8>":~4!@3S<|1vB3T33<=D{.1vUC57&` ¯M3:C5Z=24:">N"7&25Z!?` @ B3:UC57&`{328Z=:"w=<=CZ=2;B3B3[¦|OB328_~N&>"":?0=N"C8M3032425Z=2;B3B3[¦|OC5Z=vw=N&D=B3[¦|ξ >",28ξZ=<3, .w3.<=.BXC9M=03:"<3_;>":~4=@3ª28D 1vB3T3=<3< < ¯B32;_;N&><=C5"28Z=<3w3<=B=NqC1M303:<3_8>:~4=@3S2;D?1vB3LT @ 3<=28D ϕ (s)ϕC, G1(s)2;0328_ νζ :"":_8B=NAw=<=6C5v66vC5Z=vw=N&D=B3:";:w=<=CZ!N9C5Z=vw=N&D3B=[¦|>"28Z=<3w3<3B J < ζ = 0. ζ = ξ +···+ξ , ν ≥ 1 !#1'Rh^ h=± ! "+ , ) ! ϕ ) %-, .# % ζ !-,., ) # ! 1 ξ ν ν ν 1 ν ζν ν ϕζν (s) = ϕν ϕξ (s) . 2 P!+ *! , ,' # ( ' F62;2;6 / ν ϕζν (s) = M sζν = M M sζν |ν = M ϕξ (s) = ϕν ϕξ (s) , 7dN&T?TN&T M {s | ν = n} = M s = ϕ (s) . N"C;C86:7&03<=6 3B 2;C8T:&Z=`TU:M303=, ]7": , ! # ! OuCZ!< :"0!6Nd=Z=`c}B3:F=0=N"1C8M30325f4!25(t) Z=2;B3=B3:eD9C5Z=vw=N&D=B3:"D9>"28Z=<3w3<=B3[ ξ C\M=N&0!N"6287"0=N"6< ! # ! // QkZ3@qB3P{ξ ξ1 +···+ξν .1:&Z!:A©9"[\TZ!Nd4!T3v]3C5:C5Z!N&>S<=C8`SB=N97"28:"0=2;6v¬1:S<^] t2 e− 2 I= √ 2π −∞ ∞−it Z e− z2 2 t2 1 dz = e− 2 √ 2π −∞ u2 2 t2 du = e− 2 . 1. vC87&`7&28M32;03`CZ!vUw!N&D3B=N&@>"28Z=<3w3<=B=N η B3:"0!6NdZ=`B3:90=N"C8M3032425Z=2;B=NªCGM=N"0=N"62;7&0=N6< (a, σ) , M :"}4N9C5Z=vw=N&D3B!Nd@ >"28Z=<3w3<=B=N ξ = (η − a)/σ B3:0=6NdZ=`B3:0=N"C8M3032425Z=2;B=N9C1M=N&0=N6287&0!N"6< (0, 1) ,3.1:N=87&:U6v −∞−it fη (t) = e−ita e− (tσ)2 2 −∞ = eita− t2 σ 2 2 ! # ! / .1vC87&` ξ 0=N">"B3:U62803B3:90=NC5M30=24!28Z=28B=NB=N:7&0328_;T2 fξ (t) = Z1 eitx dx = . [0, 1] , M :"}4)N eit − 1 . it ! # ! / QkZ3@?C5Z=vw=N&D=B3:"D?>"28Z=<3w3<3B=[ ξ ]30!N"C5M=0324!28Z=28B=B3:"D?M3:q.1vNC8C5:B3vqC1M=N&0!N"6287"03:6 ϕξ (s) = ∞ X µk −µ e = e−µ esµ = e[µ(s−1)] . k! ! # ! / kQ Z3@{"<3B3:U6<=NdZ!`"B3:q0=NC5M30=24!28Z=28B3B=:"D%CZ=vw=N"D3B3:D?>25Z=<=w3<3B3[ UC}@3TNd@µ1vB3T33<@ f (t) ] t ∈ R ]^6:A©ª2;7"[\7"`|N&0=N"T37&280=<=C57&<=w32;C8TU:DR1vB3T3=<328DRB328T:"7":"03:D CZ!vUw!N&D3B3:D>25Z=<=w3<3B3[F, $p j#r *h^ b12;M3032;03[\>"B=N&@ª1vB3T33<3@ >"2;S2~C57&>28B3B3:8:1N&03;v628B37dN B=N"_8[\>N"287"C}@ªB328:7&03L< @ =N"7&25Z=`B3::"M30=24!28Z=28B3B=:"D B!N R ]\2~CZ=<°TN&T:>"[E"[ B3f< (t)"[¦Z=< >"2;S2;C87&>"2;B3B3[\2 t , . . . t, t < T:U6MZ=28TC8B3[\2 >[\M3:&Z!B@3287"C}@?B3280!N&>"2;B=C57">": ζ ,...,ζ 1 1 n n X n X k=1 j=1 f (tj − tk )ζj ζk ≥ 0 . n PI !#1' h h x SL:A|B3280¯U <=B3w3<3B!5, &' ") # % ' #!$# ! ., ) ! f (t) ' ,"! , # % ' # , , !-,"! ' # ! ' %-# ! # "+ +' , ' *% # % '& , , # ' ! ,'&# ( f (0) = 1 !$#+ # # / b12;:"A|U:~4!<=6:C87&`kB32;:"7&0=<3=N&7"25Z=`B3:"D{:"M=0324!28Z=28B=B3:C87&<M=03:">280U@3287"C}@?B32;C5Z=:A©qB3:CJ n X n X k=1 j=1 f (tj − tk )ζj ζk = = n X n nZ X k=1 j=1 Z X n X n e R ix(tj −tk ) o eix(tj −tk ) dF (x) ζj ζk = ζj ζk dF (x) = R k=1 j=1 Z X n eixtk ζk j=1 k=1 R P n = eixtj ζj dF (x) = Z n X k=1 2 eixtk ζk dF (x) ≥ 0. x I, sL1:0=6v~Z=<303v2;6t_4!2~C5`©ª2F7&2;:"032~6v 82 0= Z=:"7&!N3]U4!<=C8T3032;7&B3[\D%>N"03<=N"B37ª7"28:"0=2;6[S®:A|B32;0=N @ <=B3w3<3B=N=, !#1' h h=¢ #+.,' # ( '&( c # n = 0, ±1, . . . # , + ,.#' ' # '& # % '& R .! !$# ! # ! % # '& #+.,' # ( ' ( N X N X k=1 j=1 cn N cj−k ζj ζk ≥ 0 #' '&( , , , cn = ),+ # .! ( n ζ1 , . . . , ζN Zπ ! # einx dσ(x) , −π σ(x) #) ., , ) ! *) !-, %-# ., ! , # / O®CZ=<C5Z=vw=N&D=B=Nd@µ>25Z=<3w=<3B=N <=62;287MZ=:"7"B3:C87&` ])7&:|N&0=N"T37&280=<=C57&<=w32;C8TN&@µ1vUB=T33<@ f (t) ]!@3>?@ Z3@32;7"C}@%M3032;:""0!N&_8:>N"B3<32;6 kv03`2FMξZ=:7&B3:C87&N"B3<@ Sv03`"2 ξ ∞ −∞ ξ 1 pξ (x) = 2π , !-,"! ' # ! '& %-# ! , ) ! + #'&f (t)! ,. #'&. e−itx fξ (t)dt. −∞ QS:TN&©ª2~61:"0=6v~Z=v:""0!N&S2;B3<@>9:"S2~6tC5Z=vw=N&2, !#1'h h x 1:"0=6v~Z!Nc:"0=N&ª28B3<3@!, #, ξ Z∞ ) ! !-,.! # + # # )%-, .# % #ξ '& F ++(x) ' *%-# ! x, y #.! # ! ' F (x) # #' 1 lim F (y) − F (x) = 2π A→∞ ZA e−itx − e−ity f (t)dt . it *! , ,' # ( '& QkZ3@:M3032425Z=2;B3B3:C87&<{"v;4!2~6tC5w=<37dN&7&`!]3w37&: −A / 1 IA = 2π ZA −A ξ ξ 1 e−itx − e−ity f (t)dt = it 2π ZA Z∞ −A −∞ x?M3:">7&:"03B=:6¡<3B=7&28;0=NdZ=2:";0=N&B=<3w328B!N3,).1:Q=87":6v{M3:?7&28:032;62 Sv"<3B3"<37"`962~C57dN"6<{<3B=7&28;0=NdZ=1[ J 1 IA = 2π Z∞ h ZA −∞ −A eit(u−x) − eit(u−y) i dt dF (u) = it 1 = 2π Z∞ h ZA −∞ 1 = π eit(u−x) − e−it(u−x) − eit(u−y) + e−it(u−y) i dt dF (u) = it Z∞ h ZA −∞ sin(u − x)t sin(u − y)t i dt dF (u) = − t t 1 = π 1_1T3v0=C8N9N"B=NdZ=<=_;N<3_8>2;C87&B3:=]w=7&: 1 lim A→∞ π E 7"CA>L4N9CZ=254!v"2;7~]=w37&:1vB3T33<3@ B→∞ −∞ A(u−x) Z A(u−y) sin t i dt dF (u) . t sint dt = 1 . t −B 1 SA (u) = π :";0=N&B=<3w328B!N3,3.1:Q=;7&:6v A(u−x) Z sin t dt t A(u−y) Z∞ lim IA = A→∞ M N"T{TN"T lim SA (u)dF (u) . A→∞ −∞  1, x y, < Z!< <Z!< M3:"Z=vw=N&2~6 lim IA = A→∞ Zy dF (u) = F (y) − F (x) . ikj Y&(r! , ! ,"! '&# ! '& %-#! , ) ! )%-, .# % "+ ,"% .! #+ # #' x ZA Z∞ h ) ! -! ,..! # +# # '& )%-, # % / 1. 03<=>"242;6B=2;C5T:"Z=`"T:M3:&Z=2;_8B3[¦|M303<=62;03:">qM30=<=628B32;B3<@4!:"TN"_;N&B=B3:"D7&28:032;6[F, ! # ! .1vC87&` < B32;_;N">"<=C8<=6[\2cC5Z=vw=N&D3B=[\2 >"28Z=<3w3<=B3[F]0!N"C5M=0324!28Z=28B=B3[\2cM=:SB3:0=6NdZ=`B3:6v _;N"T:"B3vCcM=N&0!N"6287"/ 0=N"6< (a ξ, σ )ξ< p(a , σ ) , M :}4N9CZ=vw=N"D3B=Nd@>"25Z!<3w3<3B=N ξ + ξ 7AN"T©ª21B3:0=6NdZ!`"B3:0=N"C8M3032A@ 4!28Z=28B=N9C1M=N"0=N"62;7&0=N6< (a + a , σ + σ ) , QS2;D=C57&><37&28Z=`"B3:!] 1 1 1 2 1 2 2 2 1 2 1 2 2 2 fξ1 +ξ2 (t) = fξ1 (t)+fξ2 (t) = =eita1 − 2 t2 σ1 2 eita2 − 2 t2 σ2 2 = eit(a1 +a2 )− 2 +σ2 ) t2 (σ1 2 2 PP ξ2 ! # ! s®v66N4!>"vU|<3B3:6<=C525Z=<=w3<3B CG:~4!<=B=N&T:">[6// M=N"0=N"62;7&03:6 p @3>&Z3@32;7"C}@<3B3:U6<=NdZ=`B3:0=N"C8M3032425Z=2;B3B3:"D%C5Z=vw=N&D3B=:"D>25Z=<=w3<3B3:D^, K C8N"6:U6/425Z=2] < ξ1 ! # ! s®v66N14!>vU|9B=28_;N">"<=C8<=6[¦|qM3vNC8C8:"B3:>C5T=<|C5Z=vw=N&D3B=[¦|9>25Z=<=w3<3B ξ < ξ @3>&Z3@32;7"C}@M3vN?@ C8:"B3:>C5T:DCZ=vw=N"D3/ B3:D>25Z=<3w=<3B3:"D, F62;2;6 ¬FN"Tµ6[ >"<425Z=<µϕ0!N&B3`S(s)2"]!=2;C5Z=ϕ< ξ(x)ϕ< ξ (x) ]0=N=C5M3e03254!25Z!28B3e[ 0=N">"B3:U=628e03B=:qB=N [0, 1]. <%B32;_;N">"<=C8<=6[F]!7": F < F C8vUª2;C87&>"2;B3B3:0=N&_;B3[\2k1vB3T33<3<, !#1'%U$pUpo9(#&r M N&T3:6 1vB3T=3<3DX0=N"C8M303254!25Z=2;B3<@":U6¡<|µ|N&0=N"T37&280=<=C57&F< @ w32~C5T3<3|1vB3T33<=Dq<=6282;7"C}@q>"_;N"<=6B3:1:~4!B3:_8B=NAw=B3:"2\C5::"7&>287"C87&>"<=2", E ;0=N&B3<=w3<=6C}@9_4!2~C5`k1:"0!6v;Z!<303:">T:"D7&28Q: @ 032~6[F]<|T:"7&:03:"D?CZ=254!v"2;7A]Uw=7&D: =87&:qC5::"7&>287"C87&>"<=2 B3217&:"Z=`"T:S>_;N"<=6B3::~4!B3:"_;B=NAw3B3:2"]B3:9<>"_~N&<=6B=:ªB32;M303A2 @ 03[\>B3:"2, !#1'h^ h xz7&28:032;6NGB32;M303280=[\>"B3:UC57&"28Z=<3w3<=B 7&2;:"032~6NB=28M3032;03[\>"B=:C57"< 1:"0!6v;Z!<303v"2;7"C}@tCZ!24!v > ª<=6 :"0=N"_8:6-J !#1'Rh^ h!³^ ) '&( ∞ X ϕr (s) = n p(r) n s , r = 1, 2, . . . , #'&(* #+.,' # ( '&( .# ! ' ' ! ."+, & ) ! n=0 #'& .! "+, &# ϕ(s) = ∞ X pn sn , ) ! # #+.,' # ( '& n=0 # *+ +' ,' *% # % ' ! .# lim p(r) r→∞ n {pn } / '&") # % ' ! ! , + n = pn , s ∈ [0, 1) ! # ! / KL:C8M3:&Z=`_8v2;6C}@{7&28:032;6:DµB328M=032803[\>B3:C87&<{4=Z=@%4!:"TN"_;N&7"25Z=`UC57&>UN9M3032425Z=`B3:"Dµ7"28:"0=2;6[ .1vUN"C8C8:"B=N=, lim ϕr (s) = ϕ(s) . r→∞ .1vUC57&` .1:N=87&:U6v an = npn → a , M :}4N spn + 1 − pn n a n a n n an n = s +1− = 1+ (s − 1) → ea(s−1) . n n n lim Cnm n→∞ a m am −a an n−m n = e . 1− n n m! D 6 3$3%(' % !8 6 ,.%0/1$&%0) N&B=282"[¦Z!N4!:"TN"_;N&B!Nq<3B37&2;80=N&Z=`"B!Nd@7&2;:"032~6NµvN&>0=N7@ kN"MZ!N"C;Nv7&>280U©4N?>cªNd@^]w37&:{0=NC5M30=24!2A@ Z=2;B3<329w3<=C5Z!NvUC5M328|:> µ >{C}|2~62 SL2;03B3v~Z3Z= S<=6 %-83,9$ !8 PV M303254!25Z!`"B3[6 C5>:"D=C87&>":U6 Zx nµ − Mµ o 1 u2 lim P √ ≤x → √ e− 2 du . n→∞ Dµ 2π .103254!25Z=`B3:"D{1vB3T33<=28D@3>&Z3@32;7"C}@?1vB3T3=<@?B=:"0=6N&Z=`"B3:8:ª0=N"C8M3032425Z=2;B3<@{>"<4N −∞ T:"7&:0=Nd@{C5>"@3_;N&B!NªCG<3B37"2880!NdZ=:U6 k N"MZ!N"C;N 1 Φ(x) = √ 2π Zx e− u2 2 du , −∞ 1 Φ0 (x) = √ 2π Zx e− u2 2 du C5::"7&B=:"S2;B3<32~6 Φ(x) = + Φ (x) , £®7&:7k032;_8v~Z=`d7dN&7G@3>&Z=@3287"C}@9w=N"C87&B3[6/C5Z=vw=N&2~6:"S2;D #' !-, ( .! #+ # ( ' # ! # ,.1vUC57&` ξ ] ]B328_~N&><=C5<=6[F] M ξ = a ] Dξ = σ ,3.1:"Z=:A©q<=6 S = ξ + · · · + ξ <{>">24!2~6 n = 1, . . . 1 2 n =1'Oh^-*´^ n 2 n ζn = 0 < σ2 < ∞ n 1 n Sn −a Sn − an √ = n√ . σ n σ n # '& ! n→∞ P{ζn ≤ x} → Φ(x) !-,. # ! & ' ' # ( x ∈ (−∞, ∞) / 0=NC8C;6:"7&032;7&`M=:CZ!24!:>N&7"25Z=`Q@ *! , ,' # ( ' µ / :A©ª2;6]®M=C8w303<3<7dN&=;7&7&:`!6 ]!w=M=7&:: CMZ!24!ξ :>=N&7"025]=Z=<3`B=B3NA:w=C87&2ª` 6:A©qB3:B32[¦<=Z=_;:6"28[° B3:UC57&` B=<Z!N"C8`t[F,\s®: Z!N"C8B3:7&2;:"032~62 B32;M303280=[\>"B3:UC57&<4!:C87dN&7&:w3B3:ª4!:"TN"_;N&7"`=]w37&: {ξn0 = ξn − a}∞ n=1 .1:&Z!:A©90 n0 n > n0 x I,-+;gU OuC5Z=<9M3:C5Z=24:">N"7&25Z!`"B3:UC57&`G1vUB=T33<3D0=N"C8M303254!25Z=2;B3<@ C}|:~4<37"C}@ST9B32;M303280=[\>"B3:D1vB3T3=<3<ª0!N"C @ M303254!25Z!28B3<@ F (x) ])7&:C}|:~4!<=6:UC57&` T 0=N">"B3:U62803B=N=,QS:C87dN&7&:w3B3:qM3:TN"_;N"7&`=]3w=7&:v?TNd©4!:"D%7":"wL@ T3<µ0!N"C5ª<3032;B3B3:"D%w=<=CZ=:>":D%M30U@=6:"D <=62;287"C}@µ:T3032~C57&B3:UC57&`!]=>T:7&:"03:DµC}|:~4!B3:62;03B=N3,.1vC87&` < ,3bFN&D4!2;7"C}@ 7dN&T:2"]w37&:4=Z=@?>UC525| >"[\M3:"Z=BF@ > 7C}@B32;0=N&>28B=C87&>N −∞ < x < y < ∞ Fn (x) − ε < F (x) < Fn (x) + ε , Fn (y) − ε < F (y) < Fn (y) + ε . PU M :"}4NS4=Z3@{>C825| t ∈ [x, y] <=6282~6 F (x) − ε < F (x) 6 F (t) 6 F (y) < F (y) + ε , F (x) − ε < Fn (x) 6 Fn (t) 6 Fn (y) < F (y) + ε . E 7"CA>L4N9CZ=254!v"2;7A]3w37&:ª4=Z3@{>C825| n > n0 < t ∈ [x, y] QkZ=@"2~C5T:"B=28w3B3:v;4NdZ!28B3B3:D7":"w3T3< |F (t) − Fn (t)| < 2ε . x = −∞ <=6282~6 0 6 F (t) 6 F (y) < F (y) + ε , .1:N=87&:U6v4=Z3@{>C528| t ∈ (−∞, y] M30=< 0 6 Fn (t) 6 Fn (y) < F (y) + ε . n > n0 >"2;03B3:9B3280!N&>"2;B=C57">": |F (t) − Fn (t)| < ε . ¥ B=NdZ!:"8<=w3B3:90=N"C;C86N"7&03<3>UN&287C}@CZ=vw=N"D403v8:"D{2;C5T:B328w3B=:ªv;4N&Z=28B3B=:"D?7&:"w3T=< y = ∞ , 12;B37&0=N&Z=`"B!Nd@tM30=24!28Z=`"B=N&@t7&2;:"032~6N<=62;287/62;C87&:X7dN&T©2?M=03< B=28T:"7&:03[¦|vC5Z=:"><@|/<4!Z3@ M30=:"<3_A@ >:&Z=`B3:"D¡M3:UCZ=254!:">UN&7&28Z=`"B=:C57"<¡B328_~N&>"25Z=<=w3<3B^, sL1:"0!6v;Z!<303v"2~6 :~4!B3v<3_OM3:~4!:"B3:8: 03:~4)NS7&2;:"032~6¦, .1vUC57&`7&28M32;03` ] ]!B32;_;N">"<=C8<=6[ <<=6"2 > 7B32ª:A@3_;N"7&25Z=`B3:q:~4!B3:<7&:©ª2ª0=N"C8M3032425Z=2;B3<32", KL>"254!2;6t:":"_8B!NAw328B3<3@ ξ k = 1, . . . k < M ξk = ak , Dξk = b2k , M |ξk − ak |3 = c3k Ak = n X k=1 ak , Bn2 = n X k=1 b2k , Cn3 = n X c3k . k=1 !#1' h *^ x M 2;:"032~6N 1 @3M3vB3:>NU, )%-, # .# % ! # % Cn/Bn → 0 # ' ck lim P ξ1 , ξ2 , . . . # ,. $# a k , bk nξ + · · · + ξ − A o 1 n n ≤ x = Φ(x) . Bn c :"03:S:<3_8>2;C57"B3:"DR<Z3ZC>GC57&0=N"3<32;DOC;1:"0=6v~Z=<30=:">N"B3B3:Dµ7&28:032;6[ @3>&Z3@32;7"C}@0=NC8C828<3>UN&B3<32N&037"<Z3Z=2A@ 03<=D=C5T3<3|tC5B=N"0@4:">M=03< C87&0328Z=`"2M3:O328Z=<^,bFN7"0=N&2;T37&:"0= M3:&Z=2;7dNRC8B=N&0U@4N%4!28D=C87&>"v287R":&Z!`"S:2T:&Z=:XB328_~N&>"!]M303<3w32~6y>&Z=<@3B=<32%TNd©4!:";:/FN&T37&:0=NXC8N6:OM=:C5282%B328>25Z=<=TU:!,®£®7&:/:"7?@ TZ!:"B32;B3<@R>{T:&Z!<3w32;C87&>"29_~N&0U@4N3]:7&TZ=:B328B=<@R>{>"2~C529:>"Z!Nd©qB3:UC57&":_4!vU|UN=]=B=N"M30=N">&Z=2;B3<32k"2;7&0=NB!N0=N"_8B3[¦|%>[C5:"7dN&|?UN&B3<3@v;4!<3><37&25Z!`"B3:q7&:w3B3:9:"M=<=C5[\>UN&287C}@B3:"0!6NdZ=`B3[6_~N&T:"B3:U6¦, b1vU©qB3:RB32_~N&"[\>UN&7&`!]^w37&:O32;B37&0=N&Z=`"B=N&@M303254!25Z=`B=Nd@7"28:"0=2;6NµB=24!:TN&_8[\>UN&2;7?B=:"0=6N&Z=`"B3:UC57&`0=N"CR@ C825@3B3<@ C5B=N"0@4)N3]N%4)N&287O7&:"Z=`"T:O8<3M3:7&28_;2B=:"0=6N&Z=`"B3:UC57&< B328T:7&:"03:27"28:"0=287&<3w=2;C5T:2:":C8B3:">UN&B3<=2",pKL254!` 6N"7&2;6N"7&<3w32~C5T3<32ª7&28:03<3"Z=28B3<3@|)])NqB=N6N&7&2~6N&7&<=w32;C8T3<|6:~4!28Z3@|,.1N: =87&:U6v >9M303"2;03:A@37&B3:UC57&2;D>UC528}4)NªC5Z=24!v2879_;N"":7&<37&`UC}@:97&:6¦]w37":""[¨0=N"C;C86N"7&03<3>UN&2;6N&@6:~4!28Z=` M30!N&>"<3Z=`"B3::"7"0=Nd©9NdZ!NC5::"7&>287"C87&>"v > ª282G@3>&Z!28B3<32, ! # ! x E ª<3"T=< <=_;6280=28B3<@!5,®.103<<3_~628032;B3<3<¡B32;T:"7&:03:"D>"28Z=<3w3<=B3[ 6[ M3:&Z=vw=N"2;6M303L< @ "Z=<©ª2;B3B3:2q_8B=NAw32;/ B3<32 ξ ,YsY4!25ZN&B3B=N&@O:S<3TN δ = ξ − a 6:A©287%"[\7"`µM303254C57dN&>"Z=28B!N{a >%>"<4!2C5v66[4!>"vU| :S<3:"T δ = (ξ − M ξ) + (M ξ − a) , M32;03>N&@ <3_OTU:7&:"0=[¦| ξ − M ξ B=N&_;[\>N&2;7"C}@ CZ=vw=N"D3B3:D :"ª<3"T:D^]LN >"7":"0=N&@ M ξ − a ¯ C5<=C87&2;6N"7&<3w32~C5T:"D :S<3TU:D^,^c:"03:S<3262;7&:~4![ <3_~628032;B3<@B3294:&Z3©qB3[ <=62;7&`RC8<=C57&2~6N&7&<=w32;C8TU:D:"ª<3"T=<^, SLv;4!2;6 M3:"Z!N&5N"7&` ,suZ=vw=N"D3B=N&@9:S<3TN δ <=6282;7kB3v~Z=2;>":"2G6N&7&2~6N&7&<=w32;C8TU:2:A©q<4N"B3<32"] M δ = 0 ,.1vC57"` Dδ = σ ,"QkZ3@ Mξ = a n→∞ 2 # v628B3`S2;B3<@ =;7&:"D :"S<="T3<M=03:"<3_;>":~4=@37 B32;_;N&><=C5"25Z!<3w3<3B3[ a C80324!B=282kN&03 7O_;N:"32;B3T3v ξ1 + · · · + ξ n . n QS:"M=vC5TN"2;6v >M=03< =87&:U6 M3:";0328ªB3:C87&`%6:A©9B=:?:328B=<37&`C9M3:U6:"ª`Q> 328B=7&0=NdZ!`"B3:DOM=0324!28Z=`"B=:"DX7&28:032;6[F, s®:" ZN"C5B=: =87&:Dk7&2;:"032~62LC8v66N :~4!<3B=N"T:">":\0!N"C5M=0324!28Z=28B=B3[¦|FB328_~N&>"25Z=<3w=<3B C M ξ = a < Dξ = σ > 0ξ N"+·C8<=·6·+ξ M37&:7&<3w32~C5T3< B3:"0=6N&Z=`"B=:0=N"C8M303254!25Z=2;B=NCM=N&0=N6287&0!N"6< (an, σ√n) , .1Q: =;7&:6v ba M303<{":"Z=`"ª<| n NC5<=6M=7&:"7&<=w32;C8T3<{B3:"0=6N&Z=`"B!NCGM=N"0=N"62;7&0=N6< (a, σ/√n) < n 1 2 n n 1 P{|b a − a| ≤ ε} ≈ √ 2π QS<=C8M3280=C8<@ D(b a − a) M3:";0328ªB3:C87&9C80=N&>B328B=<3B3:"D D(b a − a) = σ2 Dξ1 + · · · + Dξn = . n2 n 8&/;/!) %-$38 % 1$ 3,.%-$323:;,923%0: 323% 8 232 DD #$ # Y"jUpr{r¡&()=n&()' .1vUC57&` ξ = (ξ , . . . , ξ ) C5Z=vw=N&D=B3[\D>28T37":"0{< t = (t , . . . , t ) ∈ R , $p j#r *!* cN"0=N&T37"2803<=C87&<3w32~C5TNd@ 1vB3T=3<@ f (t) CZ!vUw!N&D3B3:8:/>28T37&:0=N ξ : M3032425Z3@32;7"C}@ C5::"7&B=:"S2;B3<32~6 f (t) = M e = Me . .12;0328w3:"D=C87&>NS|UN"0=N&T=7&2803UC525| t ∈ R "B3:U62803B3:B32;M303280=[\>"B=NM3: t , *! , ,' # ( ' E :"_;B=NAw3<=6 w3280=28_ A C5:"[\7&<32 A = {|ξ | ≤ x, k = 1, . . . , n} <µM=vC57"` h ∈ R , E 32;B3<=6 M303<=0=N&ª28B3<32 / 1 n 1 n n ξ i(t,ξ) ξ n ξ ξ i Pn k=1 tk ξk ξ ξ ξ n k |fξ (t + h) − fξ (t)| = |M ei(t,ξ) ei(h,ξ) − 1 | ≤ M |ei(h,ξ) − 1| = }4!2 |h| = Pn k=1 |hk | < = M |ei(h,ξ) − 1|IA + M |ei(h,ξ) − 1|IĀ ≤ M |(h, ξ)|IA + 2M IĀ ≤ ≤ x|h| + 2P{ξ ∈ / [−x, x]n }, [−x, x]n ¯qM=0@=6:v8:&Z!`"B3<3T?> Rn J {t = (t1 , . . . , tn ) ∈ Rn : |tk | ≤ x, k = 1, . . . , n} . .1vC87&`4N"B3: ε > 0 ,!K®[\28032~6 x 7AN"T^]=w=7&:"[¨>[\M3:&Z=B3@Z=:C8`B=280=N">"28B!C57&>: P{ξ ∈/ [−x, x] } < ε/4 , M :}4NqM303< <=6282~6 |f (t + h) − f (t)| < ε , |h| < ε/(2x) ±p O®CZ=< ξ , . . . , ξ ¯B=28_;N">"<=C8<=6[\21CZ=vw=N"D3B3[\2F>"28Z=<3w3<=B3[ < ζ = ξ + · · · + ξ ]37&: n ξ 1 ξ n 1 fζ (t) = n Y k=1 fξk (t) . n PU CZ=vw=N"D3B3:8:>"28T=7&:"0=N ]37":9|UN&0!N&T37&2A@ ^ O®CZ=<%<3_;>"2~C57&B=Nª|N&0=N"T37&280=<=C57&<=w32;C8TN&@1vB3T33<3@ 03"2;T37&:0=N (ξ , f. . .(t), ξ ) 6:A©qB3:M3:&Z=vw3<=7&`=])ξM3=:"Z=(ξ:A©q,<3. >. .>, C8ξ2 )t ])_~N<=C8TZC> @ w32;B3<32~6 t , . . . , t ]0=N&>B3[6<{B3v~CZ >S, ¢ OuCZ=< η = ξC ¯qZ=<=B328D3B=:"21M3032;:""0!N&_8:>N"B3<32cCF6N"7&03<3=28D C =k c k ] α = 1, . . . , m β = 1, . . . , k ]7&: ξ 1 k1 k1 k kj & αβ }4!2 fη (t) = fξ (tC 0 ) , ¯C8:"M30U@©ª2;B3B=N&@?C C 6N"7&03<3!N3, C0 n kj *! , ,' # ( '& / fη (t) = M ei(t,η) = M ei(t,ξC) = M eit(C ξ0) = M ei(tC )ξ 0 = M ei(tC ,ξ) = fξ (tC 0 ) , _54!2;C8` ξ :"":_8B=NAw!N&287S>"28T=7&:"0L@ C57":&Z=28^, ! # ! / OuC5Z=< m = k ]A:"M30=24!28Z=<37&28Z=` det C 6N"7&03<3=[ C :7;Z!<3w328B:"7cB3v~Z3@S<9CZ=vw=N"D3B=N&@k>"25Z!<3w3<3B=N <=6282;7qMZ=:"7"B3:C87&` p (x) ]!7&: η = ξC <=62;2879MZ=:7&B3:UC57&` ξ ξ pη (y) = *! , ,' # ( '& K C8N6:6X425Z=2]4=Z3@?ZC> :";: / P{ξ ∈ A} = Z Z pξ (x)dx = A 1 pξ yC −1 . det C A ∈ Bk pξ yC −1 {y=xC;x∈A} <=62;2;6 1 dy = | det C | Z = ! #! / OuC5Z=< pη (y)dy = P{η ∈ {xC; x ∈ A}} . {y=xC;x∈A} η = ξC + b ¯N"F1<=B3B3:"2GM3032;:"0=N&_;:">N"B3<32]7&: fη (t) = ei(t,b) fξ (tC 0 ) . ]^7": <=62;287{>C829M303:"<=_8>":~4B3[\29M3:"0U@4!TN k UC525| ^ OuCZ=[\M3:&Z!B@F> 7"C}@{C5::"7&B3:S2;B3<@ k 1 1 k n n ξ 1 n ∂ k fξ (0, . . . , 0) = ik M ξ1k1 . . . ξnkn . ∂tk11 . . . ∂tknn < fξ (t) = k X iα X αn 1 tα 1 . . . tn M ξ1k1 . . . ξnkn + o |t|k α ! . . . α ! n =α 1 M30=< |t | + · · · + |t | → 0 , s®>:"D=C87&>":aF4!:"TN&_;[\>N"287"C}@7dN&T?©2"]TN&T?<{>:~4B3:62;03B3:6 CZ=vw=N"2", ¯RC5Z=vw=N&D=B3[\DX>"28T=7&:"0C9328Z=:"w=<=CZ=2;B3B3[6<C5Z=vw=N&D3B=[-@ $p jUpr *!±p .1vC87&` 6< T:U6M3:"B32;B37dN"6< ":~4=@3Nd@¡1vUB=T33<@ :M303254!25Z3@32;7"C}@?C8:":7&B3:"ª28B3<=2;6 ϕ (s , . . . , s ) α=0 1 α1 +···+αn k 1 α ζ 1 r 1 r r ϕζ (s1 , . . . , sr ) = M sξ11 . . . sξrr = GN&0=N"T37&280=<=C57&<=w32;C8TN&@R1vB3T33<3@ ª28B3<32~6 X αr 1 pα sα 1 . . . sr . ":~4=@3Nd@1vB3T33<3@ α fζ (t1 , . . . , tr ) fζ (t1 , . . . , tr ) = ϕζ (eit1 , . . . , eitr ) . ϕζ (s1 , . . . , sr ) C5>"@3_;N&B=[ C5::"7&B3:Q@ UP $p j#r * ^ suZ=vw=N&D=B3[\D/>"2;T37&:"0 ζ = (ξ , . . . , ξ ) B=N"_8[\>UN&287C}@OM3:"Z=<3B3:U6<=NdZ!`"B3:%0=NC5M30=24!2A@ Z=2;B3B3[6¦]32~CZ=< 1 P{ξ = α} = QkZ3@?M3:&Z=<=B3:6<=N&Z=`"B=:"8:0=NC5M303254!25Z!28B3<@ r n! r p α1 . . . p α r , p1 + · · · + p r = 1 . α1 ! . . . α r ! 1 ϕζ (s1 , . . . , sr ) = p1 s1 + · · · + pr sr . ! # ! / 1. vC87&` ξ , . . . , ξ ¯B3:"0!6NdZ=`B3:90=N"C8M3032425Z=2;B3B3[\21B328_~N&><=C5<=6[\2FC5Z=vw=N&D3B=[\2G>25Z=<=w3<3B3[ C < Dξ = b , M :"}4N =a 1 M ξα n α α k α fξ1 ...ξk (t1 , . . . , tk ) = e k P i α=1 tα aα − 21 k P α=1 bα t2α . .1vUC57&`?CZ=vw=N"D3B3[\Dµ>28T37&:0 <=62;287B328M30=2803[\>B3v >MZ!:"7&B3:UC57&` |UN"0=N&T=7&2803¨1vUB=T33 fξ (t)= (ξ∈ L, . ,. .M ,:"ξ}4)NkMZ=:"7"B3:C87&` p (x) >":C;C57dN&B!N&>&Z!<3>N"28p7"C}(x) @M3:1|UN"0=N&T37"2803<=C87&UN&B3<3@ kv03`2 1 ξ n ξ 1 ξ ξ pξ (x) = 1 (2π)n Z e−i(t,x) fξ (t)dt . S:"0!6v;ZN9:""0!N&S2;B3<@>9:"S2~6tC5Z=vw=N&2G@3>&Z=@3287"C}@?:"":"ª28B3<32~6C5>:"2;8::~4!B3:62;03B3:8:9N&B=N&Z=:"5N=, =1'/h^-*)h^ ) ' ( ∆ = {x : a < x "<32~6 =87&:Dµ7&28:032;6[°@3>"Z3@3287C}@µvU7">"280U©4!2;B3<32k:>"_;N"<=6B3:q:~4!B=:"_8B!NAw3B3:6 C5::"7&>287"C87&>"<=<625©4!v |UN"0=N&T=7&2803"242;6_54!2;C8`928S2F:~4<3BT=03<37&2;03<3D%B32;_;N&><=C5"28Z=<3w3<3B, QkZ=@B32;_;N&><=C5B3:";:ªC5>:"D=C87&>NS6N&7&2~6N&7&<3w=2;C5T:8:S:A©q<4)N&B3<@^,"QS:C87dN&7&:w3B3:C87&` CZ!24!v2879<3_c4!:"TN"_;N&B=B3:"D?1:"0=6v~Z=[¨:""0!N&S2;B3<@XxzC86¦,=7&28:032;6v{Ia5, ¬FN"T<{>9:~4!B3:U62803B3:U6tC5Z=vw=N&21M303<=>"242;6 7&:&Z=`T:1:0=6v~Z=<303:>"T3v ' # !$# #! # ! '& , =1'h^-**^ ' ") # % '& #+.,' # ( ' ( ") # ! ) ! !-,..! # +# # k=1 "+ , ( !#! ' ! ) ! !-,.! # + # # ! , + ' *% ! # #.! # ! ' ! # + # ( ) ! # #"+ +' ,'&*% # % ' ! F (x) t #+.,' # ( '&( '&.#'&'&.) & , !-,"! '&# ! ' %-#! ) ! f (t) "+ ,.( ! !$#+ # ) f (t) # #!$# ! ) ' *% ! # t = 0 / ! ' f (t) #'&( , ! ,"! '&# ! '& %-#! , ) ! ! , !$#+ # # F (x) / F (n) (x) (n) DB 8&/;/!) %-$38&/!% 8&/1$&) 18&/!% $ !: $&%(' % & %-8323% (o9# YUp!{!#1'j!'3"$p j#r V"g $p jUpr*¢^ E M=0324!28Z=<=6B=28>"[\0=:A©ª428B3B3:2E328B=7&03<303:>N"B3B3:"2 6B3:";:62;03B3:"2B3:"0=6N&Z=`"B=:"2 0=NC5M30=24!28Z=28B3<=21C5Z=vw=N&D=B3:";:ª>28T37&:0=N ξ = (ξ , . . . , ξ ) TN"T0=NC5M30=24!28Z=28B3<=21C1M3Z=:"7&B=:C57"`Q> 1 n p |A| − 12 Q(x) , x ∈ Rn e (2π)n/2 pξ (x) = }4!2 Q(x) = x A x = P a x x ] |A | ¯:M303254!25Z=<=7&25Z=`qM=:&Z=:A©q<37&28Z=`"B=::M303254!25Z=2;B3B3:D{6N"7&03<3=[ k a k , !42;C5` x :"":_8B=NAw!N&2879>28T37&:0L@ C57&:"Z="2;^,QkZ3@%32;B37&03<=03:">UN&B3B3:8:0!N"C5M=0324!28Z=28B=<@µB3<©2k"v;4!2;7q4!:TN"_;N"B3:=]3w=7&: ,pN"C5M=0324!28Z=28B=<32CZ!vUw!N&D3B3:8:%>28T37&:0=N ]}4!2 ¯OM3:C87&:A@3B3B3[\D/>28T37&:0^]7dN"T©ª2v;4!2;6 Mξ = 0 B=N"_8[\>UN&7&`B328>[\03:A©4!28B=B3[6B3:0=6NdZ!`"B3[6t0=N"C8M3032ξ4+25Z=2;aB3<32;6¦, a ∈ R bFN"D4!2~6/|UN"0=N&T=7&28031vB3T=3F< > CZ=vw=N"D3B3:8:9>"2;T37&:"0!N ξ ,=.1:TNd©ª2~6¦]w37&: n ij i j i,j i,j=1 n fξ (t) = e− 2 t M t , 1 }4!2 M = A 2;C57"`?6N"7&03<3=N=]^:"0=N&7"B=Nd@µT A ]^C8:">"M!NA4NQ> ªN&@C96N&7&03<=328D M30=24!28Z=28B3<3@ ξ J m =Mξξ . K C;N"6:U6X4!28Z=2"] −1 ij p Z∞ |A| fξ (t) = (2π)n/2 k mij k >7&:"03[¦|6:U628B37":">?0=N"CR@ i j ... Z∞ 1 eitx − 2 x A x dx1 . . . dxn LK [\"2;032;6R:"0=7&:";:"B=N&Z=`"B3v >¡6N&7"03<33v 7AN"T^]&w37":""[ "[¦Z!N1[4!=,!CsY4!28Z!N&2~6/_;N628B3vqM32;032;62;B3B3[¦|]M3:"Z=:A©q<3> x = y C < , M :"}4)N t =vC 1 −∞ −∞ n |A | = |E | = n Y µk , k=1 1 1 itx0 − x A x0 = ivy 0 − y E y 0 = 2 2 n n X 1X =i vk yk − µk yk2 2 .1:N=87&:U6v k=1 p n Z∞ 2 |A| Y 1 fξ (t) = eivk yk − 2 µk yk dyk = n/2 (2π) k=1−∞ = p |A| n Y x I3, ++~ k=1 e − 21 2 vk µk 1 = e− 2 v E −1 v0 = k=1 1 = e− 2 t C E −1 C t0 1 = e− 2 t A −1 0 t s 403v8:"DRC57&:03:"B3[F])7dN&TRTN&T>UC526:628B=7&[C5Z=vw=N&D3B=:"8:>"2;T37&:"0!N ξ C8vUª2;C87&>"v > 7A])7&:?>:T3032~C57&B3:UC57&28B=C87&>": t=0 E 7"CA>L4N9CZ=254!v"2;7A]3w37&: 1 fξ (t) = 1 − t A−1 t0 + o |t|2 = 2 ] , i2 =1 + it (M ξ)0 + t M t0 + 0 |t|2 . 2 . x I3, +AI" 1_Y4!:"TN"_;N&B=B3:";:M>"[\ξ 7&=2;T0N&28A7 :"w32;=B3`cM>UNd©qB3:2uC5>:"D=C87&>":cB3:"0=6N&Z=`"B=[¦|10!N"C5M=0324!28Z=28B=<3D J ! #'& ! , ( ! ,.!$#+ # #") .# ! ' !-, ξ = (ξ1 , . . . , ξn ) # ,. '&*),+ , ' (! ' "),+ ,*# ! "),+ , ! #'& ! ! ! # ρ(ξi , ξj ) = 0 .! # i 6= j −1 / 3V + 7dN&T©2¦4!<=N?@ *! , ,' # ( ' QS2;D=C57">"<37&28Z=`"B=:=]&2~CZ=< ¯F4!<=N"8:B=NdZ=`B=Nd@96N&7&0=<3=N3]7&:S6N"7&03<3=N 8:B=NdZ!`"B=N&@9< f (x)/ 0=N&>B=NkM30=:"<3_;>"2428B3 |UMN"0=N&T37"2803<=C87&<3w32~C5T3<3|1vUB=T33<3DTU:U6M3:"B=28B37ªA>"28=T=7&M:"0=N ξ , E "0=N"7&B3:=] 2;C5Z=< ξ , . . . , ξ B32;_;N">"<=C8<=6[F]7&: A 4!<=N"8:B=NdZ=`B=Nk<]3_8B=NAw=<37A]4!<=N"8:"B!NdZ=`B=NS6N"7&03<3=N M. .1vUC57&` B=28>"[\0=:A©ª428B3B3[\D32;B37&03<30=:">N"B3B3[\DB3:"0!6NdZ=`B3:c0!N"C5M=0324!28Z=28B=B3[\DC5Z=vw=N&D=B3[\D >"2;T37&:0?C1MZ!:"7&ξB3=:UC5(ξ7&N` > , . .p. ,(x)ξ ) −1 ξ 1 n 1 n ξ pξ (x) = p |A| − 12 xAx0 e , x ∈ Rn . (2π)n/2 KL[\"2;032;6X:037&:";:"B=N&Z=`"B=vI>¨6N&7&03<=3v 7dN"T3v >S]w37&: @3>&Z3@32;7"C}@94!<=N"8:"B!NdZ=`B3:"D6N"7&03<332;D^,K=87&:U6CZ!vN@ w=N"2"]TN&Tt6[ >"<4!28Z=<^]T:6M3:B328B=7&[ BCZ!vUw!N&D3B3:8:µ>28BT37"AB :"0=N ζ = ξB B3:0=6NdZ!`"B3:R0=N"C8M3032425Z=2;B3[ CB3v~Z=28>[6 6N&7"2;6N&7"<3w32;C8T3<=6 :A©q<4N"B3<32;6t<{B328_~N&>""<4!287&`!]B=N&D4!287"C}@?B328>[\03:A©4!28BF@ B=N&@ (n × n)@ 6N"7&03<3=N C 28T37&:0 η ]^T:U6M3:"ξB3=2;B3ζB 7&[yT:"7&:03:";:?B=28_;N">"<=C8<=6[ 28T37&:0 ξ <=62;2879M303254C57dN&>"Z=28B=<32 ξ = ηC + a, }4!2ªTU:U6M3:"B=28B37&[ C5Z=vw=N&D3B=:"8:>"28T=7&:"0=N η B=:"0=6N&Z=`"B3:0=N"C8M3032425Z=2;B3[CkM=N"0=N"62;7&0=N6< ¯qB=28>"[\0=:A©ª428B3B=N&@ (n × n)@ 6N&7&03<==N9< a ¯qM=:C57":A@3B3B3[\D?>"28T=7&:"0^]=0=N&>B3[\D M ξ , C $p j#r* ^ NC5M30=24!28Z=28B3<=2 ξ = (ξ , . . . , ξ ) C1MZ=:7&B3:UC57&`N> (0, 1) <B328_~N&>"N"287"C}@FC812;03<3w32~C5T3<4 fξ (t) = e− σ2 2 (t,t) sL1280=<3w32;C8T3N&0=<=N&B37"B3:R:7&B3:UC5<37&28Z=`"B=::"0=7&:";:"B=N&Z=`"B3[¦|/M30=28:"0=N&_;:">UN&B3<3D ]37dN&T{TN&T η = ξC fη (t) = fξ (tC 0 ) = e− σ2 2 . (tC 0 ,tC 0 ) = e− σ2 2 (t,t) . !1Up!#1'j!{#'3"$p jUprG rpn¡djlm'n .1vUC57&` ξ = (ξ , . . . , ξ ) ¯ r@ 62803B=[\DµCZ=vw=N"D3B3[\D?>"28T=7&:"0^,!.1:&Z=:A©q<=6 1 r ηi = r X c`i ξ` + ai , i = 1, . . . , m, < Z=<{>9>"2;T37&:03B3:"D{1:"0!62 x I,-+;H η = ξC +a, }4!2 η = (η , . . . , η ) ] a = (a , . . . , a ) < C =k c k ¯ (r × m)@ 6N&7&0=<3=N3,#!4!2;C8`{>"28T=7&:"0 a N"03<=N&=<3:"B3B=vI> 6N&7&0=<33v k (ξ , ξ ) k , =1'Oh^-*!± ' ,' ! C ##' #'& !-,..#'& / x I,-+8P D[η] = C D[ξ]C . *! , ,' # ( ' 1_qx I,-+;HLC5Z=24!v287A]3w37": η − M η = P c (ξ − M ξ ) < / `=1 1 m 1 m 1 ij r i i (ηi − M ηi )(ηj − M ηj ) = i r X k,`=1 r `=1 `i ` ` c`i ckj (ξ` − M ξ` )(ξk − M ξk ) . j V I KL[\w3<=C5Z3@@:79:""2;<|w=NC57&2;D?M=:CZ!24!B32;8:90=N">"28B!C57&>UNª6N"7&2;6N"7&<3w32~C5T:"21:A©q<4N"B3<32"]M=:&Z=vw=N"2;6 X X (ξ , ξ ) = c (ξ , ξ )c = c (ξ , ξ )c , r i j r ki ` k ik `j `,k=1 k ` `j `,k=1 } 4!2 c = c . $p jUpr* ^ µB3:";:6280=B3[6 B=:"0=6N&Z=`"B3[6 0=N"C8M303254!25Z=2;B3<32~6 C5Z=vw=N&D=B3[¦|>"28Z=<3w3<=BOB!N&_8:Q@ >2;6 =0 N"C8M303254!25Z=2;B3<32ª>"28T=7&:"0=N η = (η , . . . , η ) ])}4!2k>"2;T37&:0 η :M303254!25Z=2;Bµ1:"0=m6v~Z=:Dx I3, +~H]NT:6M=:"B32;B37&[ C5Z=vw=N&D=B3:";:F>"28T=7&:"0=N ξ B328_~N&>":"D=C87&>Nq6B=:"8:U62803B=:"8:qB=:"0=6N&Z=`"B3:8:0!N"C5M=0324!28Z=28B=<@^,!.1vUC57&`>28T37":"0 0=N"C8M3032425Z=2;BB3:0=6NdZ!`"B3:!, F6282;-6 J η = (η , . . . , η +, + # ! # ! ,.!$#+ # # ! ! +,.' η ' ! , ( # # Dη > 0 / I3, ) %-, .#! '& ! ζ # .! # +'&, * + # ζ = Aη # ),+ # A '&, , ,.' ! ,"# ##' ") # ! # ! , ( # !-,..! # +# # # / H=, , # , ! , ζ = a η + · · · + a η ! #' .# ! ' !-, η ##' ! , ( # !-,. .! #+ # ## ## Dζ > 0 / £®7"<µC5>":D=C57">NB328M=:C50=24C87&>"2;B3B3:CZ!24!v > 7q<=_k:"M30=24!28Z=28B3<3@?<%7&:8:FN"T37dN3]w37&:C5v66NqB32;_;N&><=C5"28Z=<3w3<=B?B=:"0=6N&Z=`"B3:0=NC5M303254!25Z!28B=N=, !#1' h^-* ' ") # % '& ) %-, .# ! ' ! η = (η , . . . , η ) ! , ( !-,..! # ik ki m 1 1 r 1 m i i 1 1 n n + # # # # *+ & +'&,'&*% # % ' , !-,"! ' # ! '& %-# ! , ) ! 1 m)%-, ") .#! '& ! , η # , + x I3, +AV" '& .# ! ' ! # B =k b k #' ! % , # ' ! ,' # ( ! # + # # , ' ! , / *! , ,' # ( '& bFN&D!4 2~6|UN&0!N&T37&2;03<=C87&<3w32~C5T3v > 1vB3T=3B=:"0=6N&Z=`"B3:/0!N"C5M=0324!28Z=28B=B3:";:tCZ=vw=N"D3B3:8: / ),+ # , 1 fη (t) = eita − 2 tBt , a = (a1 , . . . , am ) (m × m) αβ >28T37&:0=N3,=s®: Z!N"C8B3:?x HUa"L<:M3032425Z=2;B3°|UN"0=N&T=7&2803""25Z!<3w3<3B ]M3:"Z=vw=N&2~6 ξ ,...,ξ `=1 k=1 k=1 1 r m r m o o Y n X n X tk ck` ξ` = M exp i fη (t) = exp i tk a k · = exp(ita0 ) r Y exp `=1 = exp(ita0 ) exp M N&T{TN&T b k1 k2 = 7&:9:TU:B3w=N"7&25Z=`B3:M3:&Z=vw=N"2;6 k=1 `=1 k=1 (ηk1 , ηk2 ) = n n − r X − 1 2 m 1 X 2 tk ck` k=1 m X k1 ,k2 =1 c k1 ` c k2 ` , t k1 t k2 2 o r X = c k1 ` c k2 ` `=1 xzC86¦,=7&28:032;6v{I, H&P ), `=1 n 1 fη (t) = exp ita0 − 2 m X k1 ,k2 =1 o o b k1 k2 t k1 t k2 . . "V H .1vUC57&`\7&2;M3280=`ªx I3, +AV"<=62;287 62~C57&:!, M :"}4NTN&©4Nd@1T:6M3:B328B37dN η <=6282;7\|UN"0=N&T37"2803<=C87&<3w32~C5T3v >t1vB3TF@ 3 ><4N k bkk 2 2 t 7&:2~C57&`B3:"0!6NdZ=`B3:q0=N"C8M303254!25Z=2;B=NqC M η = a < Dη = b ,=Qª, NdZ!282Fv;4!:"B3282S"v;4!287qM=28032;D37&<µTµ=28B37&0=<303:?@ >N"B3B3:U6v>"2;T37&:"0=v η − a ,)s®:A|0!N&B@@:":_8B=NAw32;B3<@^]"v;4!2~6 C8w3<37dN&7"`=]w37&: M η = 0 ,3.1:UC5T:&Z!`"T3vqT:"B=28w3B3[¨>C82 6:62;B37&[ ]=7&:TU:B328w=B3[¨"[\w=<=CZ=<=7&`=]!M3:&Z=`_8v"@=C5` 0=N">"28B!C57&>:M6 η ∂ f (η , η ) = M η η = − (0) = b . ∂t ∂t M N"T3<=6 :"0=N&_;:6¦] B =k b k @3>&Z=@3287"C}@?TU:>N"03<=N&=<3:"B3B=:"D6N"7&03<332;DµCZ=vw=N"D3B3:8:>"28T=7&:"0=N η , QkZ3@tCZ > :"D @ 6N"7&03<3=[ S =k s k CZ!vUw!N&D3B3[\D >"2;T37&:"0 ζ = ηS <=62;287OB3v~Z=28>:"2?6N&7&2~6N @ 7&<3w=2;C5T:2ª:A©q<4N"B3<32(mM×ζ m) ]T:">N"03<=N"3<3:"B=B3v > 6N&7&0=<33v k (ζ , ζ ) k= S BS <µ|UN"0=N&T=7&2803 =0 1vB3T33 . .1vC87&` S ¯7dN&TN&@:"0=7&:";:"B=N&Z=`"B=N&@6N&7&0=f<3(t) =N3]==w37&f: (tS ) = e fηk (t) = eitak − k k k kk k 2 α α β 2 α β α β α η αβ β αβ αβ α ζ η d11 0 0 d22 S BS = ....... β − 21 t S 0 BS t0 ... ... = D , dαα ≥ 0 . ........ . . . dmm M :}4N M3:N8= 7&:U6v ζ , . . . , ξ B32;_;N">"<=C8<=6[F]M303<3w=2;6f 4=(t)Z3@=7&e28| α ]4=Z3@¡T:"7":", 03[¦| d > 0 C5Z=vw=N&D3B=[\2%>"25Z!<3w3<3B3[ ζ B3:0=6NdZ=`B3:0=N"C8M3032425Z=2;B3[ CGM=N"0=N"62;7&0=N6< (0, √d ) ]=NM303< d = 0 C1>2803:A@37&B=:C57"`Q> 1 <=6282~6 ζ = 0 , uO C5Z=<0=N"B3 r 6N&7&03<=3[ B 628B3`S2 m ]"7&:k6:A©2;6XC5w=<37dN&7&`!]&w37&: d 6= 0 4!Z3@ α = 1, . . . , r < d ,K 8= 7&:U6¨C5Z=vw=N&2 P{ζ = · · · = ζ = 0} = 1 ])7&:{2;C87&`%CZ=vw=N"D3B3[\29>"28Z=<3w3<=B3[ η , . . . , η= ··· = d =0 − 21 ζ 1 Pm α=1 dαα t2α m αα αα α αα C5:UC503254!:"7":"w32;B3[ B=NM30=:C57"0=N&B=C87&>"2]:"M30=24!28Z3@32;6:U6t0=N">"2;B=C57&>UN"6< mm r+1 m X M :}4N α αα r+1,r+1 m 1 m sαβ ηβ = 0 , α = r + 1, . . . , k . β=1 ηi = r X ski ζk , i = 1, . . . , m . √ ">"254!2;6 @ 6N"7&03<33v C T:U6M3:"B32;B37dN"6< ,suZ=vw=N&D3B=[\2 B328_~N&>""28Z=<3w3<=B3[ k=1 √ ξα = ζα / dαα ξα αβ αα βα η =ξC, 7&:q2~C57&` ηi = r X cαi ξα , i = 1, . . . , m . =1'Rh *!¢ r = m ! # + # ' # ( det C 6= 0 # &' r # ! # ! , ( # !-,..! # +# # # #' # ! + # , '& &#!$# ! . # # ) ' '&( ! , !$#+ # # ##' . + η 1 / .! #+ # ## , √ p (x , . . . , x ) = e (2π) det B ),+ # B =k b k # ! + # , (r × r) ,' ! *, # b = (η , η ) # x = (x , . . . , x ) # a = (a , . . . , a ) # P # ! ., + !-,'& % , ! *, # ! #'& ! ' ! ' ,. ' ,' ! ) a = M η Q(x) = b x x # ! , & ' ) & ! . , ' ! # kb k B/ k=1 η k` k ∗ k` k 1 ∗ k` k ` r 1 2 Q(x−a) r/2 k` k ` 1 r 1 r V&P *! , ,' # ( '& .GZ=:"7&B=:C57"`0!N"C5M=0324!28Z=28B=<@?C5Z=vw=N&D3B=:"8:>"2;T37&:0=N ξ <3_F7"28:"0=2;6[ I3, HUVª<=6282;7><4 / r P M N&T{TN&T 1 x2k 1 2 pξ (x1 , . . . , xr ) = √ e k=1 . ( 2π)r M3032;:"0=N&_;:">N"B3<32c>"<4N det C 6= 0 yi = gi (x1 , . . . , xr ) = r X ci` x` + ai i = 1, . . . , m, @3>"Z3@3287"C}@X>"_~N&<=6B=:":~4!B3:_8B=NAw=B3[6¡"2;B =8Z=2;62;B37&[°6N&7&03<=3[F]3:"0=N&7"B3:"D?T C , M :}4N k=1 ,p.1vUC57&` xi = Pr ∗ `=1 ci` (y` − a` ) ]p}4!2 c∗i` ¯ √ 1 pη (y1 , . . . , yr ) = |J|−1 pξ (g −1 (y)) = ( 2π)−r |J|−1 e 2 Q(y−a) , }4!2 M N&T{TN&T < det C Q(y − a) = r X r X r X x2k = c∗k` (y` − a` ) 2 x ¯24!<=B3<3w3B=N&@{6N"7&03<3!NL<=62;2;6 D[ξ] = E k=1 k=1 `=1 = r X (y`1 − a`1 )(y`2 − a`2 ) c∗k`1 c∗k`2 ,suZ=24:">N"7&25Z!`"B3:!] B = C D[ξ]C = C C `1 ,`2 =1 k b∗`1 `2 k= B −1 = C −1 (C 0 )−1 =k r X k=1 k=1 c∗k`1 c∗k`2 k =B N det B < B=N b ]UM=:&Z=vw3<=6tv7&>"2;0©4!2;B3<32G7&28:032;6[F, c c Yj 't3!#1' j#&l^1 '(r#"r#Go r##'^!()X#'3"$p jUp#o &j#lm'n#po (U#3==( QS:TNd©ª2~6 _4!2~C5`OM3032425Z=`B3v > 7&2;:"032~6vO4=Z3@ C8v66 B328_~N&><=C5<=6[¦|/:~4!<=B=N&T:">:µ0=NC5M303254!25Z!28B3B3[¦|C5Z=vI@ w=N"D3B3[¦|>28T37":"03:>=, !#1'¨h^-* ) '&( #+.,' # ( ' (# .,.. "+ ,"! ! , !$#+ # # ) %-, # ! '& ! ξξ , =ξ , (ξ. . . , . . . , ξ ) M ξ = a ! # % ! , ! , .,"% ζ = ξ + · · · + ξ / ' "),+ , ) ! !-,..! #+ # # )%-, ") & # ! '& ! , (ξ , ξ ) = b / det B = | det C|2 ,3N628B@@ r X √ |J| 1 Pr ∗ ∗ k=1 k`1 k`2 nβ αβ n 2 n nα ∗ `1 `2 n1 nk 1 nm m n ζn0 = ζn − na √ n , "+' ! ! , ( ) ! ! ,.!$#+ # # ) # ,' # , '& %-# ! + , # , ' ! # ! ., ! , B =k b k / *! , ,' # ( '& E ":"_;B=NAw3<=6 w328032;_ f (t) |N&0=N"T37&280=<=C57&<=w32;C8T3v > 1vUB=T33 C5=Z vw=N&D3B=:"8:{>28T37&:0=N ξe = / αβ ξn − a ,3.1:UC5T:&Z!`"T3v M ξen = 0 .1:N=87&:U6vqM303 :6 t M30=< < M ξenα ξenβ = bαβ ]=7&:qM3:qC5>:"D=C87&>"v9|N&0=N"T37&280=<=C57&<=w32;C8TU:D?1vB3T=3<3< k t 1 1 X f √ =1− bαβ tα tβ + o . 2n n n α,β=1 n→∞ h Pk 1 t in fζn0 (t) = f √ → e− 2 α,β=1 bαβ tα tβ , n :7"CR>L4N<{<3_17&2;:"032~6[°I3, HHªC5Z=24!v287ª4!:TN"_8[\>UN&2;6:2 v7&>280U©4!28B3<=2", K¦N&©9B=[6w=NC57&B3[6 CZ!vUw!N&2;6 =87":"D?7&28:032;6[¨@3>&Z3@32;7"C}@?C5Z=24vI>cS2;2FvU7">"280U©4!2;B3<32 n VV =1'qh^-* ) '&( ##' , ( # !-,..! #+ # ## '&,.- .# ! ' ' & "+ p = (pξ =, . (ξ. . ,,p. .). ,"ξ') + # ( ' , / ,.!$#+ # ## # ! '&n ! , (ξ−np)/√n ! n → ∞ , "+' ! ! , ( ) ) # !$#+ ,.' ! # ! ., ! , k δ p − # ),+ # δ ! #! # ! , p p k / *! , ,' # ( ' suZ!vUw!N&D3B3[\D>"28T=7&:"0 ξ M30324)C57dN&><=6X>S><4!2cC8v66[ η + · · · + η B=28_;N">"<=C8<=6[¦|9>"28T=7&:"03:> / 1 1 k k αβ α α β αβ =] }42 ]=2~CZ=<%M=03< @ 6 <=C8M3[\7dN&B3<3<M303:"<=_8:"ª25Z%<=C}|U:~4 < >qM303:7&<3>":M3:?@ Z=:A©qB3:U6tCZ!vUw!N&2",!s®:>62~C5η7&B3[\D{=_~1N&T:"B0=N"C8M3032α425Z=2;B3<@?C5Z=vw=N&D=B3[¦|>25Z=<3w=<3B η <β η η !< 6=28287q0 ><4 1 ηα = (ηα1 , . . . , ηαk ) n αβ αβ αβ αγ P{ηαβ = 1, ηαγ = 1} = 0, P{ηαβ = 1, ηαγ = 0} = pβ , P{ηαβ = 0, ηαγ = 1} = pγ , P{ηαβ = 0, ηαγ = 0} = 1 − pβ − pγ W w=<37&[\>N&@^]w37&: < 7&032;"v2;6:"21v7&>280UM©4!η28B3<=2"=, p (η αβ D β = pβ δβγ − pβ pγ αβ , ηαγ ) β 6= γ . ]#>OC5<Z!v/M30324[L4!vS28D 7&28:032;6[ M3:&Z=vw=N"2;6 !: 3$&%('*% %-8 2 6 : 4 6 1838 %:) 83/;/!)+%-$ 8 ) 8&/1$&) 18 ) $ !: &$ %('*% %-8323%0) jUpr .1vUC57&` ¯9C;12803<3w=2;C5T=<B3:"0!6NdZ=`B3:"20!N"C5M=0324!28Z=28B=<32", M :}4N ] ]"@3>"Z3@F> 7C}@ B3:0=6NdZ=`B3:10=N"ξC8=M303(ξ254!25,Z=. 2;. B3. ,B3ξ[6) p<C¦M=N"0=N"62;7&0=N6< (0, 1) B32;_;N&><=C5"i28Z==<3w31,<3.B!.N".6, k<^,".1:"Z=:A©q<=6 < χ = ξ + · · · + ξ ,pbFN"D4!2;6 1vB3T=3F< > 0=NC5M30=24!28Z=28B3<3@"28Z=<3w3<=B3[ χ J χ2 1 2 k 2 1 k 2 n i 2 1 k 2 k k 1 P{χk ≤ r} = (2π)k/2 .1:Q=;7&:6v .1:C87&:A@3B3B=N&@ Z e − 21 x x0 dx1 . . . dxk = ck ck B=Nd|U:~4!<37"C}@?<3_1vC5Z=:"><@ pχk (x) = ck rk−1 e− Z∞ Zr 1 2 tk−1 e− 2 t dt |x|≤r F6282~6 r2 2 , x ≥ 0. pχk (x)dx = 1 . ck Z∞ 2 x k−1 − x2 e dx = ck 2 Z∞ k 2 x k−1 −x2 e dx = ck 2 k 2 −1 k .1:Q=;7&:6v }#'3"$p =ck 2 2 −1 Z∞ k k x 2 −1 e−x dx = ck 2 2 −1 Γ xk−1 k 2 −1 .GZ=:7&B3:C87&`90=N"C8M303254!25Z=2;B3<@{CZ=vw=N"D3B3:"D?>"28Z=<3w3<=B3[ 2 x k−2 2 e−x dx = pχk (x) = Z∞ Γ χ2k k 2 e− x2 2 x I,-+;U k = 1. 2 , x ≥ 0. <4 x I,-+Aa" suZ=vw=N"D3B=Nd@?>"28Z=<3w3<3B!NC1M3Z=:"7&B=:C57"`Q> x I3, +da&®B=N&_;[\>N"287"C}@ χ @ ! , !$#+ # # C k C57"28M32;B@=6<µC5>:"":~4[F, k √ 1 x 2 −1 1 − x Γ e 2 , x ≥ 0. pχ2k (x) = √ pχk ( x) = k 2 x 2 22 2 V" '3"$p j#rRiS Y#' .1vUC57&`1CZ!vUw!N&D3B3[\2¦>25Z=<3w=<3B3[ ξ , ξ , . . . , ξ B32;_;N&><=C5"28Z=<3w3<3B!N (0, 1) 1 k τk = s ξ0 1 k k P ξi2 √ ξ0 k = χk B=N"_8[\>UN&287C}@9:7&B3:S28B=<32;6 s®7"`Q>L4!2;B37dNªC k C87&28M32;B@=6<{C5>":":~4![F,UbFN&D42;61vB3T3=¨0=NC5M30=24!28Z=28B3<3@<M3Z=:"7 @ B3:UC57&` τ J ZZ ZZ x I,-+;U P v e dudv , S (x) = P{τ ≤ x} = (u)P (v)dudv = a i=1 k k k ξ0 2 √ k χk u v ≤x }4!2 u v ≤x v>0 sY4!28Z!N&2~6_~N"628B=vM3280=2;628B=B3[¦| ] k−1 − 12 ( uk +v 2 ) k v>0 ak = √ u = yz v = z 1 1 . k −1 2πk 2 2 Γ( k2 ) ,W w=<37&[\>N&@^]3w37&: ∂(u, v) = z, ∂(y, z) M3:"Z=vw=N&2~6 (2.18) = ak .1:UCZ=2G_;N628B3[ Zx dy −∞ (2.19) = ak .1:N=87&:U6vqMZ=:"7"B3:C87&` Zx −∞ − k+1 y2 2 +1 dy k pτk (x) 0=N">"B=N Z∞ z k e− z2 2 r dz . y2 +1 k 2 k − w2 w e x I,-+;U 2 ( yk +1) w=z M3:"Z=vw=N&2~6 Z∞ dw = ak 2 k−1 2 k + 1 Γ 2 Zx −∞ k+1 1 Γ k+1 x2 − 2 2 pτk (x) = √ 1 + , −∞ < x < ∞ . k kπ Γ k2 − k+1 y2 2 +1 dy . k x I,JI&gU suZ=vw=N"D3B=N&@O>25Z=<3w=<3B=NµCqM3Z=:"7&B=:C57"`Q>x I,JI&gGB!N&_8[\>UN&2;7"C}@O0!N"C5M=0324!28Z=28B=B3:"DXM3:Os®7"`Q>L4!2;B37&v/C k C57&2;M328B3@=6< C8>":":~4![F, M N&T{TN&T Γ(1 + x) ∼ √2πxx e 7&:9B32;7&03v;4!B3:qv"254!<37&`UC}@^]w37&:4=Z=@?TNd©4!:8: x ∈ R 1 M30=< k → ∞ . p (x) → √ e 2π N"C8M303254!25Z=2;B3<32s®7&N` >L4!28B37dN?>">25ZXN&B3 Z=<=w=N&B3<=B :UC8C52;7A]0=N&:"7dN&>S<3DµB!N?1<=0=62"]_~N&B3<=6N">"ª28D=C}@M=<3>":>N&0=2R@ B3<=2;6¦, }#'=&$## Y"jUpr F x −x 2 τk − x2 .1vUC57&` B3[F,=KL>24!2~6tCξZ!vU,w!. N&. D3. ,B3ξv >°, η>",28.Z=.<3.w3, <=ηB3v 1 p 1 q UV a ¯%B328_~N&><=C5<=6[\2 (0, 1)@ B=:"0=6N&Z=`"B3:0=N"C8M3032425Z=2;B3B3[\29CZ=vw=N"D3B3[\2>"25Z!<3w3"<4 p Γ p+q x 2 −1 2 pFpq (x) = p q p+q , x ≥ 0 . Γ p2 Γ q2 (q + px) 2 q 2 p 2 N"C8M3032425Z=2;B3<@%Cc7dN"TU:DM3Z=:"7&B=:C57"`Q> B=N"_8[\>NQ> 7"C}@ F @ 0=NC5M30=24!28Z=28B3<3@=6< S<3ª280=N=, DB D/!8 / 2 2&: % ! rG&r¨djlm#'n##o ()j#r#mpr# .1vUC57&` (Ω, U, P) 2;C87&`q>"2;03:A@37&B3:UC57&B=:"2GM303:C87&0=N"B=C57&>:=] ζ ] n = 1, . . . ]3< ζ ¯C5Z=vw=N&D=B3[\21>"25Z!<3w3<3B=N=, $p j#r *³ .1:CZ!24!:>N&7"25Z=`B3:C87&`FCZ=vw=N"D3B3[¦|q>"28Z=<3w3<3B {ζ } C}|U:~4!<37"C}@qM3:ª>"2;03:A@37&B3:UC57& :"8: ε > 0 M303< n → ∞ . P{|ζ − ζ| > ε} → 0 ! # ! .1vC87&` ¯1w=<=CZ=:1vUC5M328|:"> >1C82803<=< B=28_;N">"<=C8<=6[¦|k<=C8M3[\7dN&B3<3D9CL>"2;03:A@37&B3:UC57&N` > vUC5M328|UN >p q:~4!B3:U6<=C8M3[\7dN&/ B3<3<,!s®:"µZ!NC5B3:7&28:032;6!2 SL2;03B3v~Z3Z=< n −−→ p , $p j#r ±´ .1:C5Z=24:">N"7&25Z!`"B3:UC57&`C5Z=vw=N&D=B3[¦|µ>"28Z=<3w3<3B {ζ } C}|U:~4!<37"C}@µM=:"w37&"280=B3:"2T ]ζ =M3<3Sv7 ζ −M^−−, B→, ζ ]=2;CZ!< ζ (ω) → ζ(ω) B=N A ∈ U < P(Ω\A) = 0 , =1'Oh^-*³^ "+ ' {ζ } ! ζ #+) #' "+ '&(* # ! '&'& / / / n n P n n n µn n P n n n n V" *! , ,' # ( '& .1vC57"` / B3<=2 ζn (ω) → ζ(ω) ,QkZ3@ZC> :"8:ªM3:&Z!:A©9<=7&25Z=`B3:";: ε <=62;287962;C57":ª>TZC> w=2R@ B=N A∈U [\ {|ζk − ζ| ≤ ε} . A⊂ .1:N=87&:U6vq>"280=B3:qC5:":7&B3:S28B=<32 n k≥n Ω\A ⊃ M N&T{TN&T \[ n k≥n {|ζk − ζ| > ε} . 0 = P(Ω\A) ≥ lim P{ <3_FC8>":"D!C57&>UNª6:B3:"7&:B3B3:UC57&<%6280=[ C5Z=24!v287A]w37&: n→∞ [ k≥n {|ζk − ζ| > ε}} , lim P{|ζn − ζ| > ε} = 0 . E 0=N&7"B3:"2\v7&>280U©4!28B3<=2B328>2803B=:=,"b1:F<3_GC}|:~4=@3ª28D=C}@M3:F>"2;03:A@37&B3:UC57&N&7"25Z=`B3:C87&<C5Z=vw=N&DF@ B3[¦|>"25Z!<3w3<3B%6:A©qB3:9>"[\0=N&7&`M3:~4!M=:CZ!24!:>N&7"25Z=`B3:C87&`=]C}|:~4=@3ªv >cC}@M3:w37&<{B=N&>2803B=:"2", C}|:~4<37"C}@T%CZ!vUw!N&D3B3:D>25Z=N"7&25Z=`B3:C87&`ªC5Z=vw=N&D=B3[¦|>"28Z=<3w3<=B w3<=B32 ζ M3:?0=NC5M303254!25Z!28B3F< >ª]<Z=T}N&©ª4:"D7":"w3T2B328M30=2803[\>B3:C87&< M30=24!28Z=`"B3:D?1vB3T=3<324!2~6tC8:""[\7"<32 / ζ An = {|ζn − ζ| ≤ ε} , M N"T?TN"TM30=< ζ(ω) − ε ≤ ζn (ω) ≤ ζ(ω) + ε , < {ζn ≤ x} ⊆ {ζ ≤ x + ε} [ An {ζ ≤ x − ε} ⊆ {ζn ≤ x} [ An .1:N=87&:U6v ζ n 7&:9M30=L4N9CZ=254!v"2;7A]3w37&: ζn ω ∈ An P{ζ ≤ x − ε} − P(An ) ≤ P{ζn ≤ x} ≤ P{ζ ≤ x + ε} + P(An ) . P{ζ ≤ x − ε} ≤ lim inf P{ζn ≤ x} ≤ lim sup P{ζn ≤ x} ≤ P{ζ ≤ x + ε} OuC5Z=< x 2;C87&`97&:"w3TN9B32;M3032;03[\>"B3:UC57&< n→∞ ]37&:q<3_x I3, II"uM3:&Z!vUw!N&2;6 x I3, II" n→∞ Fζ (x) lim Fζn (x) = Fζ (x) . 1_C}|U:~4!<=6:UC57&S]p>":":"ª2ª;:">":0U@^]^B32C5Z=24!v287%C}|:~4<=6:C87&`%M3:%>"2;03:A@37&B3:UC57&<,pb1: n→∞ 2~CZ=< 5C Z!N&:9C}|U:~4!<37"C}@T?>[\03:A©4!28B=B3:6vq0=NC5M30=24!28Z=28B3S]37&:q"2;0U©ª428B3<32, !#1'Rh^ ±p! F ⇒ F F ! + #* ' *% ! # c # '& #'&( Fζn (x) ζn ζ ζ Fζ (x) = '& P ζn −−→ c / ( 0, x < c 1, x ≥ c , "V *! , ,' # ( ' M N&T/TN&T F (c + ε) → 1 < F (c − ε) → 0 ]^7&: P{c − ε < ζ ≤ c + ε} → 1 ]^7&:%2;C87&` / , P{|ζ − c| > ε} → 0 suZ=254!v > Nd@7"28:"0=2;6N9B=N6M3:7&0328v"2;7"C}@M30="B=:C57"< F (x) < ε > 0 ,=.1:&Z!:A©9N&@^]3w37&: An = A n Bn + A n B n ⊂ A n Bn + B n . M3:"Z=vw=N&2~6 E 7"CA>L4N"2;B=C57&>UN P{(ξn < x − ε) ∩ Bn } ≥ P{ξn < x − ε} − P(B n ) CZ!24!v287A]3w37&: .1:9vC5Z=:">¨7"28:"0=2;6[ E 7"CA>L4NtM=03< F (x). ]M3:N=87&:U6v Fξn (x − ε) − P(B n ) ≤ P(An ) ≤ Fn (x + ε) + P(B n ) . P(B n ) → 0 M303< n→∞ F (x − ε) 6 lim inf P{ζn 6 x} 6 lim sup P{ζn 6 x} 6 F (x + ε) . n→∞ M=:&Z=vw=N"2;6 v7&>280U©4!28B3<=2R7&2;:"032~6[F]¦7AN"T¨TN"T ε → 0 n→∞ x ¯7&:w3TNtB=28M3032;03[\>"B=:C57"<¨1vB3T33<=< '3^= =j X r /m#r#j ( - = 1 rp#m#rp' =1'h-±* ) '&( {ξn}∞ #+.,' # ( '&( # .,.. "+, ! !-,..! # +# # )%-, .# % 2# # & ,'n=1# ,.' %-# ! # + , # M ξn = a / ' "),+, ξ1 + · · · + ξ n P −−→ a n *! , ,' # ( ' 1G 2;0328_ 032;_ Sn .! n → ∞. :":"_;B=NAw3<=6|UN"0=N&T=7&2803 1vB3T33¨CZ=vw=N"D3B3:"Dq>25Z=<=w3<3B3[ ξ ]NFw32R@ ¯C5v66v ξ / + · · · +fξ(t),QS:C87AN"7&:"w=B3:k4:"TN&_~N&7&`!]Uw=7&:4=Z3@{TNd©4!:";: t 1 1 n f Sn (t) → eiat . K¡B32;T:"7&:03:"D?:"T30=2;C57"B3:C87&<{B3v~Z3@?>[\M3:&Z!B@3287"C}@?B3280!N&>"2;B=C57">": n |f (t) − 1| < 1 . 2 # g QkZ3@7dN&T3<| t :M303254!25Z=2;B=NF1vB3T33<3@ 62~C57&:9M303254C57dN">&Z=2;B3<32 `(t) = log f (t) ,U.1:Q=87":6v14=Z=@ªZC>c":8: t <ª4!:UC57dN&7":"w3B3:1:&Z=`S<| n <=62;287 t = e` n f Sn (t) = f n M N&T{TN&T n `0 (0) = 7&: e ` t n n n ` = exp t t n n . f 0 (0) = ia, `(0) = 0 , f (0) t n − `(0) o t n → e` (0)t = eita . &rpjU#po9n '3^= =j Xr m#r"j sL1:0=6v~Z=<303v2;6 _42;C5`%4!>"27&28:032;6[F,.12803>UNd@X7&28:032;6N:"%vUC5<Z=2;B3B3:U6°_;N&T:B32q":"Z=`"ª<|Ow=<=C528ZX4=Z3@ B32;_;N">"<=C8<=6[¦|0=N&>B3:90=N"C8M303254!25Z=2;B3B3[¦|C5Z=vw=N&D=B3[¦|>"28Z=<3w3<=B^, !#1'Rh^ ±± ) '&( ξ , ξ , . . . # ,. $# M ξ = 0 Dξ = σ / 1 2 k ∞ X σ2 k '&"),+, k=1 n2 k 2 k ≤ ∞, / / ξ1 + · · · + ξ n −−−→ 0 . n QS:TN&_;N"7&25Z!`C57">":1=87":"D?7&28:032;6[ :M3<30=N"287"C}@B=NSB=280=N">"28B!C57&>:ª¬1:&Z6:"8:03:">UN3]&vUC5<Z!<3>NQ> S2;8:ªB3280=N">"2;BL@ C87&>":7G128[\S2;>N3, !#1' h-± x B32;0=N&>28B=C87&>":¬1:"Z!6:";:"03:>NU, ) '&( ξ , ξ , . . . # .,.. # ' ! # % # ,'&# ,'& %-#! #" + ,# + # ! (/ ' "),+, ),+ # 1 P{ max |ζk − M ζk | ≥ x} ≤ 1≤k≤n ζk = ξ 1 + · · · + ξ k 2 Dζn , x2 / kQ Z=@B328_~N&><=C5<=6[¦|:~4<3B=N&T:>":S0!N"C5M=0324!28Z=28B=B3[¦|CZ=vw=N"D3B3[¦|>25Z=<=w3<3B{C5M30=N">"254=Z=<3>ªT303<37&2;03<3D%C}|:~4CA>L4!v", !#1' h^ ±)¢ xzvC8<Z=28B=B3[\DR_~N&T:"B":&Z!`"S<3|µw3<=C528ZX¬1:"Z!6:";:"03:>NU, ) '&( ξ , ξ , . . . # ,. "+ ,"! "!-,.! # + # # / ' ") #% '& 1 2 / / # *+ +' , ' *% # % ' )&#' ., ! #% # M ξ = a / ikj "(!r # # # ! ) ++ % ,) # ##' #'&# &' ( ! ,"! ( %& % # ξ1 + · · · + ξ n −−−→ a . n n ) # ,"! ( %& % # µn P −−→ p , n / / µn −−−→ p . n suZ!24C87&>"<32ª>"[\7&2;TN&287<3_ªT=03<37&2;03<@R¬1:"Z!6:";:"03:>N3]=7AN"TµTN&T ><=C5<=6[¨< P{ξ = 1} = p ] P{ξ = 0} = 1 − p , k k µn = ξ 1 + · · · + ξ n ]}4!2 ξ1 , . . . , ξ n B=28_;N?@ + + + + B"! #$&%('*)+%-, ) !,.%0) 1,923%0: 7/1<5:;, 1,92&:(,.2 32 M 28:"0=<@F>"280=:A@37&B3:C87&28D1<3_;vw=N&2;7¦6N"7&2;6N"7&<3w32~C5T3<=2u6:~4!25Z!"Z=28B3<=D^,;µN"7&2;6N"7&<3w32~C5TNd@FC57dN?@ 7&<=C87&<3TNG0328N&2;7 :"0=N&7"B3[\2®_~NA4NAw=< J;0!N&_80!N&N"7&[\>N"287\0=N"_5Z=<=w3B3[\2\6287&:~4[F]~T:7&:"03[\2LM3:_8>":"Z3@F> 7cM3:1C57dN&7& 7"28:"0=287&<3T:G¯S>2803:A@37&B=:C57?@ B3v >6:~4!28Z=`=, %B=:A©ª2;C87&>"::"N P^28T=7&:">!]M3:~4!Z=25©9N&S<3|TU:B37&03:"CZ >°B=N&_;[\>N"287"C}@;28B32;0=NdZ!`"B3:DC5:>":T3vM3B3:C87&Q` >ª,UµB3:A©ªA2 @ C57">":S:7&:"0=N&B=B3[¦|q:"Q P28T37&:>kB=N"_8[\>N"287"C}@q>[\":"0=TU:D^N, G1ª>[\":03T3<B=N"_8[\>UN&287C}@9:Q P^2~6:6X>-[ @ ":03T3<^,.1:>"7&:03B=Nd@/>[\":"0=TN =;7&:R>[\":03TNµC>:"_;>"0=N"S28B=<32;6¦, S®2~C5M3:>"7&:03B=N&@>"[\":03TN >"[\":03TN%"2;_ >":_8>0=N&ª28B3<@^,¦KL[\":03TNX4!:"Z3©qB=N "[\7&`M303254C57dN">"<37&28Z=`"B=:"D <°CZ!vUw!N&D3B3:D^, E Q P28T37&[828B=280=N&Z=`"B3:D¨C8:">"Q: @ T3vM3B3:UC57&<¡:M3<=C8[\>?N >c7"C}@:~4!B3<=6y<Z=< B32;C8TU:"Z=`"T=<=6<¡w3<=C5Z=:">[6< M!N&0=N6287&0=N6<^,uQS:"M3vUC57&<=6¦]®w37&::Q P^2;T37&[ 82;B3280!NdZ=`B3:"DOC5:">:"T3vM3B=:C57"U?N >c7"C}@RCZ!vUw!N&D3B3:DR>25Z=<=w3<3B3:D ξ ]^M=03<3B3<=6QN > ª28DXB=N{TNd©4!:U6 :"N P^28T=7&2 B32;TU:7&:"0=:"2Gw3<=CZ!:">":21_8B=NAw32;B3<32, sY4!28Z!N&>%>[\":03T3vµ:Q P^2~6N ]6[ :"M303254!25Z=@32;6 0U@4t_8B=NAw32;B3<3D ,p.1: =;7&:"D/M=:CZ!24!:>N&7"25Z=`B3:C87&&Z!28B3<3n21:91vB3T33<3¡<]d>":"_~6:A©qB3:=]84!03v8<=2LM!N&0=N6287&03[ CZ=vw=N"D3B3:Dª>25Z=<3w=<3B3[F,&bFN&"CZ >L4N"2;6[\2u> =;TC8M3280=<=628B37"2\_8B=NAw32;B3<@ CZ!vUw!N&D3B3:D>25Z=<3w=<3B3[ B3:C}@37{CZ=vw=N"D3B3[\D|UN&0!N&T37&2;0^,£®7&< M303<3B=<=6?N >c77&29©2ª_;B=NAw32;B3<@TN&TR< < x 0=NC5M303254!25Z!28B3[ TN"T ξ ,bFNkB3ξ<|C5Z=24v"287ªC86:"7"03287&`TN&TB=N n C5Z=vw=xN&D=B3[¦|q>"25Z!<3w3<3B^]0=NC5M30=24!28Z=28B3B=[¦|9TN"T ξξ]U> w=NC57&B3:UC57&<] M x = M ξ = m < Dx = Dξ = σ , suZ=vw=N"D3B3[\D >"2;T37&:"0 ]}4!2 B328_~N&><=C5<=6[\2?":µ0=NC5M30=24!28Z=28B3B=[\2{C5Z=vw=N&D=B3[\2 >"28Z=<3w3<=B3[ CF1vUB=T33<32;D?0!N"C5M=(x032,4!.28.Z=.28,B=x<@ ) P{x x6 x} = F (x) = F (x) ]=@3>"Z3@3287C}@{6N"7&2;6N"7&<3w32~C5T:"Dµ6:~4!28Z=Q` > B32;_;N&><=C5:~4!<=6[¦|>q0!N&>"B=[¦|vC5Z=:"><@|,!suZ=vw=N&D3B!Nd@{>"25Z!<3w3<3B=N ξ @3>"Z3@3287"C}@{M3:~4=Z!25©?N @ S2;D{:"M303254!25Z!28B3F< > C5Z=vw=N&D3B=:"D?>"25Z!<3w3<3B3:DR, %["v;4!2~6 0=NC8C86N"7&03<3>UN&7&`?7&:"Z=`"T:?M3:">7&:"03B=[\2>[\":03T3<^H, OuCZ![\":03TNC5:UC57dN&>"Z3@3287B328_;B=NAw3<37"25Z=Q` @ B3v > w=NC57&`982;B3280!NdZ=`B3:"D?C5:">:"T3vM3B=:C57"<^]7&:q0=N&_8Z=<3w3<=21628©4!vM3:>"7&:03B3:D"7&:03B3:"D>"[\:"03T:"DB32;C8Iv @ S2~C57&>28B3B=:=]3N>9M3032425Z=`B3:6 CZ=vw=N"2 =87&:90=N"_5Z=<3w=<32F<=C5w=28_;N"287A, k k k 2 k 1 n k k ξ B D ) 32 $32&%0: 16 D8 2 6 $ : 3$3%(' % -% 8 2 6 £L6M3<=03<3w32~C5TNd@%1vB3T33<@?0=N"C8M303254!25Z=2;B3<@ }4!2 Fn∗ :"M30=24!28Z3@3287"C}@%C5::"7&B=:"S2;B3<32~6 Fn∗ (x) = µn µn (x) , n ¯qw=<=CZ=: x ]3B321M30=28>":UC}|:~4=@3ª<| x ,!^N"6287"<=6¦]=w37": F @3>&Z3@32;7"C}@?C5Z=vw=N&D3B=:"D?>"25Z!<3w3<3B3:D^, =1'R*z ! , +") x *) ε > 0 . #' ! ,. #'&. ∗ n k lim P{|Fn∗ (x) − F (x)| < ε} = 1 . vUC5M328|:6¦, M :}4N *! , ,' # ( ' bFN"_8:>"2;6 C5Z=vw=N&D3B=:"2RC8:"[\7&<32 ¯ > 2803:A@37"B3:C87&`/vC8/ M325|N3] µ ¯w3<=C5Z=:tvUC5M328|:">/< F {x(x) <¯ :"x}7&B=:C5<=7&25Z=`B=Nd@ w!N"C57":"F7d(x) N/vC8=M325|UP{x :">!,16_7&x}2;:"03=2~6[ p SL2803B=v;Z=Z=<%CZ=254!v"2;7v7&>280U©4!28B=<3217&28:032;6[F, n→∞ n ∗ n k k UI T3< N"_8:"`"2~6 7&:"w3TN6< −∞ = z < z < · · · < z < z = ∞ w3<=C5Z=:">v > :UC5`tB!N M=03:628©9v7 @ , M :"}4)NX:"7&B3:UC5<37"25Z=`B=Nd@w=N"C87&:"7dNXM3:M=NA4)N&B3<@ x < M=03:628©9v7&:T (z , z ] 0!N&>"B!N p = [z , z ) ,=.103<{:&Z=`S:6:"QP2;62G>"[\":03T3< p &Z=<=_8T3"280=:A@37&B3:C87~@=6 ) − F (z ) ∪m k=1 ∗ Fn (zk+1 k 1 m k+1 ∗ n k m+1 i ∗ k pk = F (zk+1 ) − F (zk ) = k ∗ k k+1 zZk+1 p(x)dx , }4!2 p(x) ¯ M3Z=:"7&B=:C57"`0=N"C8M303254!25Z=2;B3<@ C5Z=vw=N&D3B=:"D¡>"28Z=<3w3<=B3[ ξ ,LbFNd4 TN&©4![6 M303:U625©qv7&T:6 (z , z ] C87&03:"<=7"C}@M30U@=6:"v8:"Z=`"B=<3T^]^MZ!:"ªNd4!`µT:"7":"03:8:%M303:M3:"0=3<3:"B!NdZ=`B=N ,.1:C87&03:"2;B3B=N&@X1<3;v0=N%B=N"_8[\>N"287"C}@ ;<=C57&:80=N66:"D^, <=C87&:";0=N"66Nª4N"287qM30=24C87AN">&Z=2;B3<32F:M3Z=:"7&B=:C57"<%0=N"pC8M3032425Z=2;B3<@{MZ=:"7"B3:C87&< p(x) ]!B3:w3v>Q@ C87&>"<37"25Z=`B=NT?0!N&_8<328B=F< >S, E =28B3TNGB328<=_8>"2~C57&B=:"D91vUB=T33<3<0=N"C8M303254!25Z=2;B3<@q6:A©ª2;7c[\7&`FM3:&Z!vUw=28B=NcB=N1:C8B3:">UN&B3<=N=, !#1'* h^ F (x) #.! # ! ,*#'& ! n → ∞ zk k ∗ k √ P{ n( sup |Fn∗ (x) − F (x)|) < z} → K(z) = −∞0 k=−∞ .! . ! O®CZ=< ¯0328ª28B3<32Fv0=N">"B32;B3<@ B32;0=N&>28B=C87&>": zb α 1 − K(b zα ) = α ]37&:C1>2803:A@37"B3:C87&`Q> 1−α k+1 4=Z3@>UC528| x >"[\M3:"Z=B@32;7"C}@ zbα zbα Fn∗ (x) − √ < F (x) < Fn∗ (x) + √ . n n suZ=254!v > Nd@ 7"28:"0=2;6NXv7&>"2;0U©ª4)N&287A]Yw=7&:C{>"2;03:A@37&B3:UC57&`N> + =~6M3<303<=w32;C8TN&@1vB3T=3<@0=N"C8M3032425Z=2;B3<@ F 0=N">"B3:U62803B=:C}|U:~4!<37"C}@T F , !#1'/*^ * x Z=<=>"28B=TU:, ) ' ( F (x) ) ! !-., .! #+ # # )%-, .# % ξ F (x) ∗ n ! %-# ! , ) ! !-,.! # + # # ! # ) ( ' ,' n # ,. , + # ,+ )%-, .# % ξ "),+, .! n → ∞ /' ∗ n P{ lim B sup n→∞ −∞9>"[\:"03T3< n x = m∗1 = 1X xk , n >[\":"0=:"w3B=N&@94!<=C8M3280=C8<@{>">":~4<37"C}@C5Z=24!v >cS<=6 :"0=N&_;:6-J k=1 n s2 = α∗2 = 1X (xk − x)2 . n KL[\":03:"w=B3[\2G6:U628B37"[¨M3:0U@4!TN j :"M303254!25Z=@F> 7"C}@%C5::"7&B3:S2;B3<@=6< 1X < α = X(x − x) , m = x n KL[\":03:"w=B3[\2G6:U628B37"[¨<{32;B37&0=N&Z=`"B3[\21>[\":"0=:"w3B3[\216:62;B37&[¨@3>&Z3F@ >c7"C}@%CZ=vw=N"D3B3[6<{>"28Z=<3w3<3B!N"6<^, M 28:03287&<=w32;C8T3<32%6:62;B37&[ M ξ M3:"0U@4!TN k "v;4!2~6:":"_;B=NAw=N"7&`%w3280=28_ m ]YNR32;B37&0=N&Z=`"B3[\D 6:62;B37 M3:"0U@4!TN k v;4!2;6°:":"_;B=NAw=N"7&`%w3280=28_ α ,Kyw=N"C87&B3:UC57&<^] α 2~C57&`µ4!<=C5M=280=C8<@^]:"":_8B=NAw!N&2;6N&@ M (ξ − M ξ) 7dN&T3©ª21w3280=28_ σ , k=1 n n j k ∗ j ∗ j k=1 k j k=1 k k 2 k k 2 H =1'*^-± ! *% # #'& *+, ' .# ! ' '& ! ' .#' ' ) & '&# !$# '& %-#! # '&, .! n → ∞ / *! , ,' # ( ' 1. :"Z=:A©q<=6 / suZ=vw=N"D3B3[\2F>"28Z=<3w3<=B3[ ":"Z=`"ª<|w3<=C825Z ξ1 = xk1 , . . . , ξn = xkn . ξi B32;_;N">"<=C8<=6[¨<%:~4<3B=N&T:>":0=N"C8M3032425Z=2;B3[F] M ξ lim P 1_FC5::"7&B3:S2;B3<3D n→∞ = M ξ k = mk ,s®:" Z!NC5B3:9_~N&T:"B3v o n ξ +···+ξ 1 n − M ξi > ε = 0 . n m∗k = CZ!24!v287A]3w37&: i ξ1 + · · · + ξ n , M ξi = mk , n =1'*^ ! *% # #' ! , ( # #'& "+, '& .# ! , '&' !& '&.#'&'&.) & ' # ! #' %-#! #' , / .103254!>N"03<37&28Z=`"B=:k4!:TNd©ª2~6/Z=2~6 6v", 11'*z ) '&( f (t) = f (t , . . . , t ) #! # ! , ) ! *! ) ' # ! # # ' *% ! # (m , . . . , m ) # +.,' # ( '&( )%-, .#! '& ! (ξ , . . . , ξ ) "+' ! .# ! ' '& ! # ! '& ! ) (m , . . . , m ) / ' *),+ , #+.,' # ( '&( ) %-, .# % n → ∞ "+' .# ! ' '& ! f (m , . . . , m ) / f (ξ , . . . , ξ ) *! , ,' # ( ' .1:9:M3032425Z=2;B3 B32;M303280=[\>"B3:UC57& 0)(|ti − mi | < δ, i = 1, k ⇒ |f (t1 , . . . , tk ) − f (m1 , . . . , mk )| < ε). (n) Ai (n) = {|ξi − mi | < δ} k \ E 7"CA>L4N9CZ=254!v"2;7~]=w37&: < (n) (n) Ai i=1 (n) (n) B (n) = {|f (ξ1 , . . . , ξk ) − f (m1 , . . . , mk )| < ε}. ⊂ B (n) ⇒ B (n) ⊂ (n) P{|f (ξ1 , . . . , ξk ) − f (m1 , . . . , mk )| > ε} 6 *! , ,' # ( ' ' # ! # KL>"242;6t1vB3T=3 k [ (n) Ai . i=1 k X i=1 (n) P{|ξi − mi | > δ} → 0 . / f (y) = f (y0 , . . . , yk ) = N6287&<=6¦]=w37&: k X (−1)k−j Ckj y1k−j yj . j=0 αk = M (ξ − M ξ)k = M = k X j=0 k X j=0 (−1)k−j Ckj (M ξ)k−j ξ j = (−1)k−j Ckj (M ξ)k−j M ξ j = k X j=0 (−1)k−j Ckj m1k−j mj = f (1, m1 , . . . , mk ) < "P n α∗k = k X = M N&T{TN&T (−1)k−j Ckj xk−j j=1 P n m∗i −−→ mi k 1X 1 XX (xi − x)k = (−1)k−j Ckj xji xk−j = n i=1 n i=1 j=1 n 1 X n i=1 k X xji = (−1)k−j Ckj (m∗1 )k−j m∗j = f (1, m∗1 , . . . , m∗k ) ]<=_1Z=2;66[°CZ=254!v"2;7A]3w37&: j=1 P f (1, m∗1 , . . . , m∗k ) −−→ f (1, m1 , . . . , mk ), 7&:92;C87&` P α∗k −−→ αk . #/18 67,92&%/ 3%-8 32 .1vUC57&` ¯t6B3:A©ª2~C57&>:R0!N"C5M=0324!28Z=28B=<3D < ¯/w=<=CZ=:>N&@|UN"0=N&T37"2803<=C87&<3TN3]:~4!B=:"_8B!NAw3B3::M303254!25Z3L@ @ ~2 6Nd@M3:O0=N"C8M3F03254!25Z=2;B3ª,YK 7AN"T:6 CZ!vUw!N&2 θ B=N"_8θ[\>N"287"C}@/M=N"0=N"62;7&03:U60=NC5M303254!25Z!28B3<@^,bFN"M303<=62;0^] θ = R ,)s®::"7&>287"C87&>"<32FB=21@3>&Z3@32;7"C}@>_;N&9B=N @ |U:A©ªxdF 428B3(x) <3<{1vB3T33<=< θ = θ(x , . . . , x ) :77=5Z=2~628B37&:>q>"[\:"03T3<]TU:7&:"0!Nd@{>TN"TU:U6 @ Z!<3":qC86[C5Z=2F&Z!<3_8TN9T M=N"0=N"62;7&03v θ = θ(F ) , M N"TRTN"T x ¯{B=28_;N">"<=C8<=6[\2ªC5Z=vw=N&D=B3[\2ª>25Z=<=w3<3B3[F])7&: θ ¯µCZ=vw=N"D3B=Nd@µ>25Z=<=w3<3B=N=] :M3032425Z3@32~6Nd@X0=N"C8M3032425Z=2;B3<32;6 F , 1> UNd@/1vUB=T33<@/:"7 =5Z=2~628B37&:>µ>"[\:"03T=UN&2;7"C}@OC87AN"7&<=C87&<3T:"D^, M N&T3<=6X:"0=N"_8:U6¦]&:"32;B3TNFM=N"0=N"62;7&0=N1@3>"Z3@3287"C}@C87AN"7&<=C87&<3T:"D^, S®Z=<3_;:C57"`kC57dN&7&N&2~6:6vªM=N&0=?N @ 62;7&03v:"2~C5M32;w3<3>N"287"C}@vCZ!:">"<3@=6<B32~C8628ª28B3B=:C57"<^]=C5:UC57&:A@37"25Z=`B3:C87&<&< =~F128T37&<=>"B3:UC57&<^, U1 U##& :"_;B=NAw=N"287:"7"C8vU7C57&><32FC5<=C87&2;6N"7&<3w32~C5T:"D{:"ª<3"T=<^]7&:q2;C87&` ∗ 1 n ξ ∗ k ξ M θ∗ = θ . K T NAw32;C87&>"2GM303[\":"0=:"w3B3:2cC80324!B=282F<{>"[\:"03:w3B3v > 4<=C5M32;0=C5S, !#1'X*^ ¢^ ) '&( (x , . . . , x ) ! ! , !-,..! # +# # F (x) ,.'&# ,' %-#! +,.# m + # ! # 1 σ2 / ' *),+ , n M x = m, Dx = Ds2 = ξ σ2 n−1 2 , M s2 = σ n n α4 − 2α22 α4 − 3α22 α4 − α22 1 −2 + =O , 2 3 n n n n # ! # %-# #' α / *! ,.,.'&# ( ' .1280=>"[\24>Nv7&>280U©4!28B=<@kZ=2;8T: M30=:">"2;0U@F> 7"C}@^, E C57dN"B3:"><=6C}@FB=N\4:"TN&_~N&7&28Z=`C87&>"2 / 4 4!>"vU|M3:UCZ=254!B3<|0=N&>28B=C87&>=, F62;2;6 n s2 = .1:Q=87":6v n 1X 1X (xi − x)2 = (xi − m + m − x)2 = n i=1 n i=1 n n n X 1X 1X 2 (xi − m)2 + (x − m)2 − (x − m) (xi − m) = (xi − m)2 − (x − m)2 . n i=1 n n i=1 i=1 n M s2 = 1X σ2 n−1 2 M (xi − m)2 − M (x − m)2 = σ 2 − = σ . n i=1 n n K®>24!2~6 yk = xk − m . UV M :}4N .1:Q=87":6v s2 = α∗2 = < n n k=1 k=1 1X 1X yk = xk − m . n n y= n n 1X 1X 2 (yk − y)2 = yk − y 2 n n k=1 k=1 1 X 2 2 2 2 X 2 yk − y = 2 yk + y 4 . n n n N"62;7&<=6¦]3w37": M yk = 0 < (α∗2 )2 n B32;_;N">"<=C8<=6[F,3.1:Q=;7&:6v>qC8v662 k=1 yk M y4 = 1 n4 k=1 n X :"7~Z=<3w3B=[ :7/B3v~Z3@¡7&:&Z=`T: n C5Z!N&5N"2;6[¦| ><4N ] ,!K¡0328_;v~Z=`d7dN&7&2]<=6282~6 M y M y y = α k 6= ` 2 2 ` k M 2 2 n h X k=1 .10324C87dN&>"<4N n 1X X 2 i 1 n(n − 1) 2 α2 = α4 + (n − 1)α22 . yk2 /n = M yk4 + 2 yk2 y`2 = α4 + 2 n n 2 y2 k=1 >><4!2 k6=` i X 1 hX 2 . y + 2 y y k ` k n2 n y2 = M :"}4)N k=1 k6=` n n n X 1 2 X 2 1 X 2 2 1 X yk2 yk = 3 yk + 2 3 yk y` y n n n .1:Q=87":6v 7AN"TTN"T M (y`1 y`2 y`3 y`4 ) `1 ,`2 ,`3 ,`4 =1 k=1 n2 M M (yk y`3 ) = h1 n y2 k=1 n X yk2 M=03< =0 k=1 M (yk y` yr2 ) i =M k6=` k=1 n 2 i 1 X yk2 = α4 + (n − 1)α22 , n , F288T:7&28M=2803`qv"254!<37&`UC}@^]w37&:!] k 6= ` 6= r k=1 Ds2 = Dα∗2 = M (α∗2 )2 − (M α∗2 )2 = α4 − α22 α4 − 2α22 α4 − 3α22 −2 + . n n2 n3 s®:" Z!NC5B3:G4!:TN&_;N"B3B3:D7"28:"0=2;62 >[\":03:"w3B=:"2C50=24!B32;2@3>"Z3F@ > 7"C}@B=2;C862;S2;B3B3:"D:"32;B3T:"D6N&7"2;6N&7"<3w3R2 @ 5C T:";::A©q<4)N&B3<@^]N9>"[\:"03:w3B=Nd@4!<=C8M3280=C8<@{@3>&Z3@32;7"C}@µC862;S28B=B3:"D%:328B3T:D^,=b12;C;628ª28B3B3:D{:"32;B3T:"D 4!<=C5M=280=C8<3&Z3@32;7"C}@?C87AN"7&<=C87&<3TN ŝ2 = n−1 2 s , n 7AN"TB!N&_8[\>UN&2~6Nd@<=C8M30=N">&Z=2;B3B=Nd@>"[\":03:"w=B=Nd@q4<=C5M32;0=C5<3@^, K¡:"ª2;6tC5Z=vw=N&2]36:A©qB3:9M3:"TN&_~N&7&`!]w37&: 1 < 1 Mα = α + O M (α − α ) = O . n n K®7":"0=N&@<3_ =87"<|?1:0=6v~Z{>2803B!N3]2;C5Z=L4NqCZ=254!v"2;7A]w37&: ∗ k ∗ k k k 2k Dα∗k = O 1 n . 2 h^1iª&33j!!" :"32;B3T3<^N, OuC57&2~C57&>28B3B=:SM3:7&0328:">UN&7&`!]&w37&:"[:"32;B3TN θ M303< Z!N"C8`T{:328B3<=>N&2~6:6vqM=N"0=N"62;7&03v, E 328B=TN θ B!N&_8[\>UN&2;7"C}@C5:UC57&:A@37"25Z=`B3:"D^]2~CZ=< ∗ n ∗ n n→∞ M=03<3&Z!<©N?@ P !#1'R* ^ ! n → ∞ M θn∗ → θ θn∗ −−→ θ . *! ,.,.'&# ( ' .1:9B3280!N&>"2;B=C57">"vBG128[\S2;>N Dθn∗ → 0 # '&*),+ , # ! , θn∗ ' ' # ( , / / K®:UC5M3:"Z=`"_;v"2~6C}@B3280=N">"2;B=C57&>:6 <3_ |θn∗ − θ| > ε C5Z=24v"287A]3w=7&: P{|θn∗ − M θn∗ | > ε} 6 Dθn∗ → 0. ε2 |θn∗ − θ| 6 |θn∗ − M θn∗ | + |M θn∗ − θ| . M N"T{TN"T (∀ε > 0)(∃n0 )(∀n > n0 ⇒ |M θn∗ − θ| 6 ε/2), |θn∗ − M θn∗ | > ε/2 ,.1:N=87&:U6v P{|θn∗ − θ| > ε} 6 P{|θn∗ − M θn∗ | > ε/2} → 0 , 7&:q2;C87&` θ −−→ θ , !#1'*^ ^ ", %-, ( # # ' !-, ( # #'& ∗ n P '&# ( & # ! ,&,"%-, ( &#' ! , ( #m' ! # % ' m α / 2k ∗ k # # ' '& .! ) &) #'&.,. α∗k k = 1, 2 . . . mk αk 2k * - - U #rp(#& # %2;03:"DR0=N"C;C528@3B3<@%@3>&Z=@3287"C}@{4!28B3B3:?>":M303:UCF:B!Nd|:A©4!2;B3<3< :"32;B3:"T{C1B=N&:9N"C8:&CZ >c7&B3:B328M30=2803[\>B3[¦|:~4!B=:6280=B3[¦|0=N"C8M303254!25Z=2;B3<3D F = {p(u, θ)} , _;N&><=C}@3S<3|:"79:~4!B3:8:9M=N&0!N"6287"0=N θ <%M=vC57"` x , . . . , x ¯>[\":"0=TNª<3_F0=NC5M30=24!28Z=28B3<3@{C1MZ!:"7&B3:UC57&`N> CGB32;<3_8>2;C87&B3[6 M=N&0=N6287&0=:6 θ , p(u, θ) K®>24!2~6tC8:">62~C57&B=Iv > M3Z=:"7&B=:C57"`0!N"C5M=0324!28Z=28B=<@ 1 f (u, θ) = 11C5B3:!]w37&: .1vC57"` n Y n p(ui , θ), u = (u1 , . . . , un ) . i=1 Z f (u, θ)du = 1 . (∗) ¯:"32;B3TNM=N&0=N6287&0!N θ ,!.1:&Z=:A©q<=6 = ϕ(x , . . . , x ) n θn∗ 1 n g(θ) == M θ∗ = Z ϕ(u)f (u, θ)du . (∗∗) n $## Y"jUpry±)h^ 1B=1:0=6N&3<=28D S<3S2;0=N C82;62;D=C57&>UNMZ!:"7&B3:UC57&2;D J(θ) = Z ∂ ln f (u, θ) 2 f (u, θ)du . ∂θ n {f (u, θ)} B!N&_8[\>UN&2;7"C}@F>"25Z!<3w3<3B=N !#1'*-³ x B32;0=N&>28B=C87&>":N":X¯ ¬10=N6280=NU, # # '& '&'&# {f (u, θ)} # # # ! , ' ,"! .$# % '& '&# ) ! , !-,..#' ., (u1 , . . . , un ) + # !$# ! .,θ'&n∗( = ϕ(x , ! 1,, . .#.' , ! x) n)θ # ' # #' #'& # !-,..#' Dθn∗ > a u = [g 0 (θ)]2 . J(θ) ,"! -! ,..# ' ., ! ! ! , +'& +) ,.#'& ' "),+ , '& (! *'&*),+ ,*#! "),+ , ) #'&.) #' % K # # ,. ## ' u1, . . . , un # θ # ' '&(*.,.. ## ' θ ' ,"! #,# % ' ∂ ln f = K(ϕ − g(θ)) . ∂θ *! ,.,.'&# ( ' s®:" ZN"C5B=:SM303254!M3:"Z=:A©ª2;B3<@=6 7&28:032;6[ / Z E 7"CR>L4N9C5Z=24!v287A]w37&: Z n f (u, θ) ∂ ln f (u, θ) du = 0 ∂θ Wc6B3:A©q<=6 M3280=>":"2G0=N">"28B!C57&>:B!N n < ∂f (u, θ) du = 0 ∂θ g (θ) = Z ϕ(u) Z ϕ(u)f (u, θ) n < g 0 (θ) = ∂f (u, θ) du . ∂θ ∂ ln f (u, θ) du . ∂θ <{>[\w37&2;6 032;_8v~Z=`d7dN&7<3_F>"7":"03:8:0=N&>28B=C87&>N n g(θ) Z ∂ ln f (u, θ) g (θ) = [ϕ(u) − g(θ)]f (u, θ) du . ∂θ .1:&Z=:A©q<=6 n < ϕ = ∂ ln f pf . sM3:U6:"S`N>°B3280=N">"2;B=C57&>UNª¬1:S<¯LSLvUB3@3TU:>C8TU:8:FM3:"Z=vw3<=6 B3∂θ280!N&>"2;B=C57">":N":d¯¬10=N"62;0=N ϕ1 = (ϕ − g) [g 0 (θ)]2 = Z (ϕ − g) p 2 ∂ ln f f du 6 ∂θ f 2 Z ∂ ln f 2 f du ∂θ Z (ϕ − g)2 f du 6 J(θ) Dθn∗ . )B=N&T{0=N">"2;B=C57&>UN>B3280!N&>"2;B=C57">"2k¬1:"ª<¯ SLvB@3T:">UC5T:";:14!:UC57&<35N"287"C}@%7&:}4N9<%7&:"Z=`"T:q7&:}4N3]3T:}4N M30=:"M3:033<3:B=NdZ=`B3[ ϕ < n 2 n n p ∂ ln f p f = K(ϕ − g(θ)) f . ∂θ $, #%-, / ! 1B=1:"0!6N&3 <@ S<3S2;0=N2;C57"`qB321w37&:q<3B3:21TN"T h M N"T%TN"T 7&: J(θ) = M ∂ ln f (x, θ) 2 ∂θ M , _4!2~C5` x = (x1 , . . . , xn ) − ∂ ln f (x, θ) ∂θ J(θ) = D = Z n CZ=vw=N"D3B=N&@>"[\:"03TN ∂ ln f f du = 0, ∂θ ∂ ln f (x, θ) ∂θ . . ϕ1 *^ M N"T{TN"T f (x, θ) = <{C5Z=vw=N&D=B3[\21>"25Z!<3w3<3B3[ n Y p(xi , θ) i=1 ∂ ln p(xi , θ) ∂θ B328_~N&>"":S0!N"C5M=0324!28Z=28B=[F]M3:Q=;7&:6v J(θ) = D }4!2 n n X ∂ ln p(xi , θ) X ∂ ln p(xi , θ) D = = nJ1 (θ) , ∂θ ∂θ i=1 i=1 J1 (θ) = Z∞ ∂ ln p(t, θ) 2 p(t, θ)dt ∂θ 2;C57"`q<3B=1:"0!6N&3<@ S<3S2;0=N:~4!B3:";:ªB!N&&ZK>L4!28B3<3@^, $, #%-, QkZ3@q4!"25Z!<3w3<3B<3B37&2;80=N&Z><3B=1:"0!6N&3<3< k<=S280!NS_~N"628B3@3287"C}@ C5v66:"D^, / E 7&B3:"ª28B3<=2 −∞ e(θn∗ ) = [g 0 (θ)]2 D(θn∗ )J(θ) B=N&_;[\>N&2;7"C}@ =~F128T37&<=>"B3:UC57&`N> :328B=T3<^, E 32;B3TN =;F128T=7&<3>"B!N >F0=NC8C;6N&7&03<=>N&2~6[¦|SM303254!M3:"Z=:A©ª2;B3<@|] TU:}4N e(θ ) = 1 ,COuCZ=< e(θ ) < 1 < e(θ ) → 1 ]7&:9;:">":0U@37ª:NC5<=6M37":"7&<3w=2;C5T:D3=~F128T37"<3>"B3:UC57&<, ¯>[\":03TNS<3_FB=:"0=6N&Z=`"B3:90=NC5M30=24!28Z=28B3B=:"D{82;B3280=N&Z=`"B=:"D{C5:">:"T3vM3B=:?@ ! # ! .1vC57"` C57&<{<{<3_8>2;/ C87&B3:"D4!<=(xC8M3,28.0!.C5.<3,2;xD σ) N&2~6[6t6N"7&2;6N"7&<3w32~C5T3<=6 :A©q<4N"B3<32~6 m ] ∗ n ∗ n 1 ∗ n n 2 (u−m)2 1 p(u, m) = √ e− 2σ2 σ 2π J1 (m) = Z∞ ∂ ln p(u, m) 2 1 p(u, m)du = √ ∂m σ 2π Z∞ 2 1 (u − m)2 − (u−m) 2σ2 e du = 2 . 4 σ σ QkZ3@tB32~C8628ª28B3B=:"D:328B3T=< 6N&7"2;6N&7"<3w32;C8T:"8:R:A©q<4N&B=<@B3280!N&>"2;B=C57">":N":%¯X¬10!N"6280!NµM303<=B3<=6N&2;7 >"<4 −∞ Dm∗ > −∞ σ2 . n OuCZ=<>OTNAw32;C87&>"2?:"32;B3T3< m >_5@37&`O>"[\:"03:w3B3:2C50=24!B32;2 .1:Q8= 7":6v:328B3TN m = x @3>&Z=@3287"C}@&=;F12;T37&<3>B3:"D^, ] 7": M m = m ]< Dm = , ¯ >"[\:"03TNO<3_µB=:"0=6N&Z=`"B3:/0!N"C5M=0324!28Z=28B=B3:"D¡;28B32;0=NdZ=`B3:"D¡C8:">":T3vML@ ! # ! .1vC57"` B3:C87&<¨CR<3/ _8>2;C87&B3[6 (x6,N".7&.2;.6, xN"7&<3) w32~C5T3<=6 :A©q<4N&B=<32;6 m < C:328B3<=>N&2~6:"D 4! Nd@ MZ=:"7"B3:C87&` ∗ ∗ 1 m∗ = x ∗ n p(u, θ) = √ J1 (θ) = √ 1 2πθ Z∞ 1 2πθ (u − m)2 2θ2 − e− (u−m)2 2θ 1 2 − (u−m)2 1 2θ e du = 2 . 2θ 2θ .1:Q=87":6v94=Z=@?B32~C862;S28B=B3:"D?:"32;B3T3<{<=62;287q62;C87&:qB3280!N&>"2;B=C57">": −∞ Dθ∗ > 2θ2 . n ∗ σ2 n NC8C;6:"7&03"25Z!<3w3<3B3[ <=62;2;6 α4 = 3σ 4 . .1:Q=87":6v M s2 = M N&T{w37": n−1 2 2(n − 1) 4 2(n − 1) 2 σ , Ds2 = σ = θ n n2 n2 xzC;6¦,37&2;:"032~6vH3, 5, n−1 < 1, n e(s2 ) = _8B=NAw3<=7~]:328B=TN θ = s B321@3>"Z3@3287"C}@3=;F12;T37&<3>B3:"D^]B3:@3>"Z3@3287"C}@%N"C8<=6M37&:7&<3w32~C5T3<&=~F128T37&<=>"B3:D^, E 328B3TN θ = sb @3>&Z3@32;7"C}@B32~C862;S28B=B3:"D{:"=28B3T:"D< ∗ ∗ 2 2 Db s2 = 2θ2 . n−1 .1:Q=87":6v7=;F12;T37&<3>B3:C87&`D=87&:D:328B=T3<%628B3`S2124!<=B3<33[ e(b s2 ) = QkZ3@:328B3T=< s20 >"[\M=:&Z=B@F>c7"C}@0!N&>"2;B=C57">N 1 Db s2 nJ 1 (θ) = n−1 < 1. n < Ds = 2θ , n 7&:92;C57"` s @3>"Z3@3287C}@B32~C8628ª28B3B=:"D?<3=~F128T37"<3>"B3:D:328B3T:D4![\":"0=TN (x , . . . , x ) ¯"<3B3:U6<=NdZ!`"B3:0=N"C8M303254!25Z=2;B=N M s20 = θ 2 2 2 1 Cc<=_8>"2~C57&B=[6 N <{M=N&0=N6287&0=:6 J1 (p) = n k k k → PN (k) = CN p (1 − p)N −k θ=p N X ∂ ln PN (i) 2 ∂p , PN (i) = N X i p − N − i 2 i i N CN p (1 − p)N −i = . 1−p p(1 − p) QkZ3@B32~C862;S28B=B3:"D?:"32;B3T3<{B32;0=N&>28B=C87&>":N&:ª¯¬10=N"62;0=NM303<3B3<4 i=0 Dp∗ > .1:&Z=:A©q<=6 p∗ = M :"}4)N 7&:92;C57"` i=0 p∗ ¯B=;F12;T37&<3>B=Nd@?:"32;B3TN3, M p∗ = p p(1 − p) . nN n x 1 X i x . = N nN i=1 < Dp∗ = p(1 − p) , nN a&g ! # ! / .1vC57"`>[\":03TN <=62;28790=N"C8M303254!25Z=2;B3<32F.1vNC8C8:"B=NCGM=N&0=N6287&0=:6 (x1 , . . . , xn ) P (k) = M :"}4)N J1 (λ) = QkZ3@B32~C862;S28B=B3:"D{:"=28B3T3< .1vC57"` λ∗ = x , M :}4N An < ∂λ λ P (i) = ∞ X i λ −1 2 λi i! e−λ = B32;0=N&>28B=C87&>":N&:ª¯¬10=N"62;0=NSM=03<3B3<=6N"287q><4 i=0 ∗ λk −λ e . k! < M λ∗ = M xi = λ i=0 Dλ∗ > Dλ∗ = λ n λ . n ,&.1:Q=87":6vF:"=28B3TN λ∗ = x 1 . λ @3>&Z3@32;7"C}@ =;F12;T37&<3>B3:"D, D :;2&) 3,./1,.2&%0: 16 8&/1$&) 18&/!:;, 4 /!$3/ 38 ) /!)+%-83,./!4 $p jUpr ±)*^ ®O CZ=N&7"25Z=`B3:C87&`\CZ=vw=N"D3B3[¦|c>25Z=<=w3<3B ζ 7dN"TU:>N=]w37&: M=03:"0U@37A]w37":M3:UCZ=254!:">UN&7&28Z=`"B3:UC57&` {ζ } ,. '&'& %-# ! ! , ( , C M=N&0=N6287&0!N"6< (A , B ) <3Z=< M30=:C57": (A , B )@ , '& ' %-# ! ! , ( , , !#1' *z´ ! # %-# '&# !$#'& %-# ! #' m # '& ! n → ∞ ! *% n→∞ n n n n n #' m , '& ' %-# ! ! , # , ! , #' ! , *! , ,' # ( '& suZ=vw=N&D3B!Nd@>25Z=<=w3<3B=N / ∗ k nm∗k = n X mk , q 2k m2k −m2k n / xki @3>"Z3@3287"C}@C8v66:"DµB32;_;N">"<=C8<=6[¦|%:~4!<3B=N"TU:>":q0=NC5M30=24!28Z=28B3B=[¦|%C5Z=vw=N&D=B3[¦|%>"25Z!<3w3<3B^, OuC5Z=<T:B328w32;Bµ7&2;:"032A@ 7&<=w32;C8T3<3D6:U628B37 m = M x ]U7&:BT =87":"DC8v66216:A©qB3:kM30=<=628B3<=7&`ª=28B37&0!NdZ=`B3v > M303254!25Z!`"B3v > 7"28:"0=2;6v", M N&T{TN&T M x = m < Dx = m − m ]37&:qCZ!vUw!N&D3B=N&@>25Z=<3w=<3B=N i=1 k i 2k k k i 2k i 2 k 2k nm∗ − nmk m∗ − m k p k qk = m2k −m2k (m2k − m2k )n NC5<=6M37":"7&<3w=2;C5T="[\:"03:w3B3[¦|F6:628B=7&:"> α 4!:"TN"_8[\>N"287"C}@kC5Z=:A©qB3282, E ;0=N&B3<=w3<=6C}@?C5Z=vw=N&2~6 k = 2 , !#1'*z! ! # %-# '&# !$#'& %-# ! #' α # '& ! n → ∞ ! *% , + # ! n ∗ k ., '&'& %-# ! ! , ( , , !-, #' !-, α , q / =s *! , ,' # ( '& .1:&Z=:A©q<=6 y = x − M x ] k = 1, . . . , n ,3.128032;M3<3ª2;6 1:"=0 6v~Z=v94!Z3@ / 4 α∗2 2 2 k k s2 = α4 −α22 n k n n k=1 k=1 1X 1X 2 (yk − y)2 = yk − y 2 , n n s2 >q><4!2 }4!2 y = (y1 + . . . , yn )/n ,!suZ=vw=N&D3B=vI> >25Z=<=w3<3B3v ζn = M303254C57dN">"<=6t>q>"<4!2 a+ ξn + η n ]3}4!2 n √ 1 X yk2 − y2 − M y12 n n k=1 n √ 1 X 2 ξn = √ (yk − M yk2 ), ηn = −y2 n. n Fb N&M=:6B3<=6¦]!w37&: y ] k = 1, . . . , n ¯B32;_;N">"<=C8<=6[\2F:~4!<3B=N"T:">":90=NC5M30=24!28Z=28B3B=[\2SCZ=vw=N"D3B3[\2F>25Z=<3w=<3B3[F, M N&T TN&T k=1 2 k M |ηn | = √ Dx1 α2 nM y2 = √ = √ , n n <3_FB32;0=N&>28B=C87&>N G128[\S2;>NSC5Z=24v"287A]3w=7&:4=Z3@ZC> :"8: P{|ηn | > ε} 6 ε>0 M |ηn | α2 = √ → 0. ε ε n E 7"AC >L4NF"M=Nd4N&2;71CM3032425Z=`B3[6X0=N"C8M3032A@ 4!28Z=28B3<=2;6 ξ ,s®: Z!N"C8B3:G328B37"0=NdZ=`B3:"D9M30=24!28Z=`"B3:D97&28:032;62C5Z=vw=N&D3B!Nd@9ζ >"25Z!<3w3<3B=N ξ NC5<=6M=7&:"7&<=w32;C8T3"[\":03:"w=B3[\D6:62;B37 α = s M303< n → ∞ N"C8<=6M37&:7&<3w32~C5T3<{B3:0=6NdZ!`"B3:0=N"C8M303254!25Z=2;B%CcM!N&0=N6287&0=N6< α , , =1' *z3h ) ' #(% '& {ξ } ,. ' '& %-# ! ! , ( !-,..! # +# # , n n n 2 2 4 2 α4 −α22 2 n 2 #+.,' # ( '&( )%-, .# % , !-, #' !-, m , σ # ),+ # M ξ = m → m # Dξ = σ → 0 ) ' ( ) ! .! #+ # #, *!+!$#'&' '&*% ! ##' '& *!+!$#'&' #!$# ! → .0 / #%-,.'& #"! g(x) "+ #'& ! ") ! + ! ,& g (m) 6= 0 ")m,+ , #+ ,'&# ( '&(* )%-, .# % η = g(ξ ) ,. '&'& %-# ! ! , ( , , ! , #' !-/ ,' g(m), |g (m)|σ # '&&#'&( ! ∞ n n=1 n m−mn σn n n n *! , ,' # ( ' .10=< / %:A©2;6tC8w3<37dN&7"`=]3w37": .103254C57dN&><=6 ηn n n→∞ .1:&Z!:A©99>"<4!2 ηn = ηn0 + δn ,3KL>"242;6 CZ=vw=N"D3B3:"21C5:"[\7&<=2 ηn = g(ξn )IAn + g(ξn )IAn = g(m) + g (m)(ξn − m)+r(ξn )(ξn − m)2 IAn + g(ξn )IAn . ]}42 ηn0 = g(m) + g 0 (m)(ξn − m) N"C;C86:"7"03<=6 δn = −g(m)IAn −g 0 (m)(ξn − m)IAn + r(ξn )(ξn − m)2 IAn + g(ξn )IAn . ηn0 − g(m) g 0 (m) ξn − m = |g 0 (m)|σn |g 0 (m)| σn n n 2 n a"I ®s : Z!N"C8B3:S7&2;:"032~62SI, PUISM3:C5Z=24:">N"7&25Z!`"B3:UC57&`9CZ=vw=N"D3B3[¦|>25Z=<=w3<3B ξn − m n ξn − m m − m n = + σn σn σn NC5<=6M37":"7&<3w=2;C5T=<¡B3:"0=6N&Z=`"B!N/C%M=N&0!N"6287"0=N"6< ,u.1:N=87&:U6vtM3:UCZ=254!:">UN&7&28Z=`"B3:UC57&` NC5<=6M=7&:"7&<=w32;C8T3< B3:0=6NdZ!`"B=NXC{M=N"0=N"62;7&0=N"6< (g(m), |g (m)|σ ) ,®(0,.1:1)TN&©ª2~6¦]w37&: δ /σ −−→ 0 , 1_%B3280!N&η>"2;B=C57">NG128[\S28>UN C5Z=24!v287A]w37&: n P n n n σ2 P(An ) 6 √ n = σn3/2 → 0 . σn .1:N=87&:U6v K√ KP(An ) 6 σn → 0 εσn ε P{|g(m)|IAn > εσn } 6 ¥ B=NdZ=:8<3w=B3:ª4!:"TN&_;[\>N"287"C}@^]Uw37&: P{|g(ξ )|I > εσ } → 0 . QªN&Z=282]UM=03<=628B3@@{B=280=N">"28B!C57&>:¬1:S<¯LS®vB@3T:>C5T:8:=]"M3:&Z=vw3 εσn } 6 6 εσn q 3/4 3/4 |g 0 (m)| M (ξn − m)2 M IAn |g 0 (m)|σn σn |g 0 (m)|σn 6 = → 0. εσn εσn ε B=N&T:"B=28^] P{IAn |r(ξn )(ξn − m)2 | > εσn } 6 2Kσn 2K mn − m KM (ξn − m)2 6 + → 0. εσn ε ε σn .1:N=87&:U6vM=:CZ!24!:>N&7"25Z=`B3:C87&`kCZ=vw=N"D3B3[¦|>25Z=<3w=<3B I3, PIkM3:UCZ=254!:">UN&7&28Z=`"B=:C57"` δn /σn C}|:~4!<37C}@9M=:ª>2803:A@37&B=:C57"<TB3v~ZC>ª,3.1:S7&2;:"032~62 ηn0 − g(m) δn ηn − g(m) = + 0 |g 0 (m)|σn |g 0 (m)|σn |g (m)|σn N C5<=6M37":"7&<3w=2;C5T="254!2;6X_54!2;C8`S1:"0=6v~Z=<=03:">T3vqC}|:A©ª2;8:Fv7&>280U©4!28B3<3@94=Z3@M3:UCZ=254!:">UN&7&28Z=`"B3:UC57&<NC5<=6M=7&:"7&<=w32;C8T3< B3:0=6NdZ!`"B3[¦|X>"2;T37&:"0=:">=,#¥ C8<=6M37&:7&<3w32~C5TNd@B3:0=6NdZ!`"B3:UC57&`µ:_8B=NAw!N&287A]pw37&: P{(ξ − m)/√n 6 x} C}|:~4!<37?@ C}@TB=:"0=6N&Z=`"B3:U6vR0!N"C5M=0324!28Z=28B=F< > CT:">N"03<=N"3<3:"B=B3:"D/6N&7"03<332;D k σ k [66N&7&2~6N&7&<=w32;C8T3<=6 :A©q<4N"B3<32~6¦, !#1'/*^ * ) '&( ξ = (ξ , . . . , ξ ), n = 1, 2, . . . # + ,'&# ( '&( ,. '& ' %-# n `k ! ! , ( .# ! ' ! *" #'&, n n1 nr M ξn` = mn` → m` (` = 1, . . . , r), (ξ n` , ξnk ) σ`k (n) (k, ` = 1, . . . , r) , n n→∞ g(x) = g(x1 , . . . , xr ) = √ ),+ # ! ) '&( ) ! .! # +# #, σ &*!+(m !$#'&−m' ) '&n*%→! 0 m =σ (m(n), → # # ' ' *! + $ ! # & ' ' # . ! # ! # % . , ' #"! "+ # .' ! ") ! + ! , / η. . .=, mg(ξ) , . . . , ξ ) .# % , σ = P g g σ #(),+# g = (m) # '&# ( ,*#2' .! n → ∞ n` ` `k `k 1 r n n1 2 nr r k,`=1 Zx 2 n √n(η − g(m)) o u 1 n P e 2 du . 6x → √ σ 2π −∞ k ` k` i ∂g ∂xi BB %-, / ' 8 / '*%-832 6 / &%-8&/ a&H U3 ¨1!1Up3!( .1vUC57&` ¯{>"[\":03TNq<3_9C52~628D=C87&>N0=N"C8M303254!25Z=2;B3<3D ] ])_~N&>""2~C57&B=:"8: r@ 628(x03B=:", 8. :. . M=, xN&0!)N"6287"0=N θ = (θ , . . . , θ ) <3_S6B3:A©2;C87&>N Θ ⊂ F (u,,θ).1vC8u7&`∈>" 2;T37&:0¯1vB3T3= g(x) = 7AN"T:">N=]w37&:9v0=N&>B328B3<=2 (g (x), . . . , g (x)) 1 n 1 1 r r r mg (θ) = t, } 4!2 t = (t , . . . , t ) ] m (θ) = m (θ), . . . , m (θ) < m < B32;M3032;03[\>"B3:t0=N&_;0328ª<=6::"7"B3:C8<37&25Z!`"B3: θ >:""Z!N"C87&< QS:"M=vC57"<=6¦]w37&:q>28T37&:0 1 r g g1 gr g = (g 1 , . . . , g r ) = n 1 X n ]):~4!B3:_8B=NAw3B=: R∞ = M gj (ξ) = −∞ gj (u)dF (u, θ) mg (Θ) θ ∈ Θ θ = m−1 g (t) gj (θ) _8B=NAw=28B3<3D ]®7":2;C87&` 1X gr (xi ) n i=1 , n g1 (xi ), . . . , M303<=B=NA4=Z!25©q<37q:""Z!N"C87&< m (Θ) , $p j#r ±!±p E 328B3T=< θ = m (g) B!N&_8[\>UN&2;7"C}@S:"=28B3T:"D]dM3:&Z=vw32;B3B3:DªM3:F62;7&:~4!vS6:U628B37&:>=, .1:~4!:"B3:8:ª03:~4Nª:"32;B3T3< θ C8:C57":A@37&25Z=`B3[F, =1'X*z±p # ! , θ = (θ , . . . , θ ) # ) %-,.# , ! ,"!&!$#% # # )! ,..# m (θ) = g i=1 g ∗ −1 g ∗ #' ' ' # ( / *! , ,' # ( ' F62;2;6 / ∗ ∗ 1 ∗ r g θk∗ = m−1 gk (g 1 , . . . , g r ) , }4!2 g −−→ m ] k = 1, . . . , r ]\w37":t@3>"Z3@3287"C}@¨:"":"ª28B3<32~6y7&28:032;6[ : C5:UC57&:A@37"25Z=`B3:C87&<=;6M=<303<3w32~C5T3<3| 6:62;B37&:>=]< m ¯ B328M30=2803[\>B3[\2F1vUB=T33<3<^,!.1:ªZ!2;662kH3,-+ P k gk −1 gk P θk∗ = m−1 −→ m−1 gk (g 1 , . . . , g r ) − gk (mg1 , . . . , mgr ) = θk . OuC5Z=TNAw=2;C57">"21vB3T3=<3D ] k = 1, . . . , r ]#>[\"0=N"7&`µ1vB3T33<=< g (x) = x ]#7&:µB!N&S2>"2;T37&:"0F@ B3:2µv0=N&>B328B3<=2µM3032;>"0=N"ªN&2;7"C}@>tC5g<=C8(x) 7&2;6v¡v0=N">"B32;B3<3D 4=Z=@:"[\w3B=[¦|6:U628B37&:>=,s®>":2%B=N&_;>N&B=<32562;7&:~4 6:62;B37&:> GM3:"Z=vw3<Z%M3:"7":6v"]=w37&:C8vU7"`q288:qC8:C57":"<37q>qM30=<30=N&>B3<3>UN&B3<3<7"28:"0=287&<3w=2;C5T=<|G<3=~6M3<303<=w32;C8T3<| 6:62;B37&:>=, ! # ! / N"C;C86:7&03<=6 C52~628D=C87&>":qM3:TN&_;N"7&25Z!`"B3[¦|q0=NC5M303254!25Z!28B3<3D%C1MZ!:"7&B3:UC57&`9>2803:A@37&B=:C57"< ] u > 0 ]!CkB=28<3_;>"2;C87&B3[6 M=N"0=N"62;7&03:6 λ ,).1:C57"03:"<=6 :"=28B3T3v4=Z3@ λ C5B!NAw=NdZ!NqCS1vUB=T33<32;D {p(u, λ)} = λe = ] 9 N ~ _ & N & 7 ~ 2 t 6 CF1vB3T3=<328D g(x) = x ,s®M30!N&>"254=Z=<3>:C5Z=24Iv >cS2;2F0=N&>28B=C87&>": g(x) = x k k −λu 2 m1 (λ) = M x1 = 1_1v0=N&>B328B3<3@ Z∞ uλe−λu du = B=N&D42;6 :"32;B3T3vM3:q62;7&:~4!v6:U628B37":"> m (λ) = x 1 . λ 1 QkZ3@?>"7&:03:"D{1vB3T33<3< g(x) = x2 λ∗1 = <=6282~6 m2 (λ) = M x21 = Z∞ 1 . x u2 λe−λu du = 2 , λ2 k da P 28ª28B3<=2;6 vU0!N&>"B=28B3<@ m2 (λ) = 1 n Pn i=1 x2i @3>&Z3@32;7"C}@:328B3TN λ∗2 = n −1/2 1 X x2i . 2n i=1 < λ @3>&Z3F@ >c7"C}@?C8:C57":A@37&25Z=`B3[6<^, λ % ¯ = B " : = 6 & N = Z " ` 3 B ? : = N 5 C 3 M 3 5 2 ! 4 5 2 ! Z 8 2 3 B = B & N @>"[\:"03TNCªB=28<3_;>"2;C87&B3[6"2428B3B3:D>[\S217&2;:"032~6[ H=, +;PC5Z=24!v287A]3w37"::"21:"=28B3T3< 0=N6< m ! # ! / .1vUC57&` < 1 σ2 N"C;C86:7&03<=6 ∗ 1 ∗ 2 n p(u, m, σ 2 ) = ] g1 (x) = x g2 (x) = x2 (x−m)2 1 √ e− 2σ2 . σ 2π ,M3:"Z=vw3<=6 CZ=254!v > ª<32Fv0=N&>B328B3<3@q4=Z3@%6287&:~4)Nq6:628B=7&:"> n m = x, σ 2 + m2 = <3_1T:"7":"03[¦|B!Nd|:~4<=6:328B3T=< 1X 2 x , n i=1 i n m∗ = x, (σ 2 )∗ = 1X 2 x − (x)2 = s2 . n i=1 i U3 #'3r 3j X O$##'3( Y=$ Y !rp .1vUC57&` >[\":"0=TNc<3_0!N"C5M=0324!28Z=28B=<@9C¦M3Z=:"7&B=:C57"`Q> ;:">":0U@^]U:7ª>28(xT37":",03.B3. :. ,8x:M=)N&0!N"6287"0=N θ ,SvB3T33<3@ 1 n L(x1 , . . . , xn , θ) = n Y ] p(u, θ) u ∈ ]_;N">"<=C}@3Nd@^]A>"::""ª2 p(xk , θ) TN&T%1vB3T3=<@ B=N&_;[\>N&2;7"C}@%1vUB=T33<32;D%M30=N&>&4!:"M=:~4!:"<@^, E 328B3T:D?B!N&<3:&Z=`S28;:9M30=N&>&4!:"M=:~4!:"<@B=N"_8[\>UN @ 2;7"C}@?C87AN"7&<=C87&<3TθN θ = θ (x , . . . , x ) 4!Z3@?T:7&:"03:D xzH=, +A L(x , . . . , x , θ ) = max L(x , . . . , x , θ) . E 32;B3TNqB=N"<3":"Z=`"ª288:qM=0=N&>&4!:"M3:~4:""<3@?:M3032425Z3@32;7"C}@7&2~6¡vC5Z=:"><32;6¦]!w37&:?>"280=:A@37&B3:C87&`M3:"Z=vw3<37&`4N"B3B3v > T:"B=T303287"B3v > >"[\":03T3v (x , . . . , x ) 4!:"Z3©qB=N"[\7&`9B=N"<3":"Z=`"ª28D^,bFNM30=N"T37&<3T2F>62~C57&: max L <3ªv796N"TC5L< @ 6v6 log L , ! # ! / b1:0=6NdZ!`"B3:2c0=NC5M30=24!28Z=28B3<=2G 0. ,SvB3T33<@{M30=N">d4!:M3:~4!:"<@_~N&M3<3ª287"C}@?>9>"<4!2 L(x1 , . . . , xn , m, σ) = (2π) −n 2 σ −n exp n − n o 1 X (xi − m)2 . 2 2σ i=1 M N&T/TN&T < 4:C57"<35N?>c7µ6N&TC5<=6v6N%M=03"7&:03:"DS4!<=F12;0328B=3<=NdZ1vB3T3=<3< >F7&:"w3T2 :7&03<3=N"7&25Z!`"B3:1:"M=0324!28Z=28B]"7&:F>1=;7&:"D97&:w3T2 4!2;D=C57&><37&28Z=`"B3:ª4!:UC57&<=5N&287C}@?6N&TC8<=6v6 =;7&:"logD%1LvB3T33<=<^, (x, s ) ! # ! / QkZ3@{"<=B3:6<=N&Z=`"B=:ª0=NC5M30=24!28Z=28B3B=:"D?>"[\":03T3<:"QP2;6N n <=62;2;6 2 }4!2 L(x1 , . . . , xn , p) = pν (1 − p)n−ν , ] p = P(xi = 1) P(xi = 0) = 1 − p ]3< ν ¯w3<=CZ!: =8Z=2;62;B37&:>q>"[\":03T3<^]0=N">"B3[¦|24!<3B=<332",!.1:Q=87":6v log L(x1 , . . . , xn , p) = ν log p + (n − ν) log(1 − p) , < ν n−ν ∂L = − . ∂p p 1−p 28ª28B3<32~6tv0=N">"B32;B3<@@3>&Z3@32;7"C}@:328B3TN p∗ = ν . n b1287"03v;4!B3:qM303:>"2;03<37&`!]w37&:9>7=87&:D7":"w3T2G4!2;D=C57&><37&28Z=`"B3:ª4!:UC57&<=5N&287C}@?6N&TC8<=6v6 1vB3T3=<3< L , ! # ! / .1vUC57&`9>"[\:"03TNª:"N P^2;6N n 0=N">"B3:U62803B=:0!N"C5M=0324!28Z=28B!NSB=N9:7&032;_8T2 [0, θ] ,=K =87":6 CZ=vw=N"2 p(u, θ) = .1:Q=;7&:6v ( θ−1 , z ∈ [0, θ], z∈ / [0, θ] . 0, M303<{>C825| i = 1, . . . , n . >M=03:"7&<=>"B3:U6tCZ!vUw!N&2 . 0, E ":_8B=NAw=<=6¡w32;0328_ , M :"}4N , . . . , x , θ) = 0 M=03< θ ∈ [0, x ) ]^< L(x , . . . , x , θ) = θ M303< θ ∈ (x , ∞)) ,=x.1:Q=;=7&:6max v6xN&TC5<=6v6 1L(x vUB=T33<3< 4!:C87&<35N"287"C}@?>7":"w3T2 , .1vUC57&` x , . . . , x ¯>[\":"0=TN<3_F0!N"C5M=0324!28Z=28B=L<@{CkMZ=:"7"B3:C87&Q` > p(u, θ)θ ]!=_;N">"x<=C}@3ª28D?:"7qM=N"0=N"62;7&0=N θ , .103<3>24!2~6t28_G4!:TN&_;N"7&25Z!`C57">Nª7&28:032;6v?:"qN"C8<=6M37&:7&<3w32~C5T:"D%B=:"0=6N&Z=`"B3:UC57&<{:328B3T= @ 4!:M3:~4!:"<@^,LSLv842;6 M30324M3:&Z!N"5N&7&`!]w37&: +AFC57&<3B=B3:"2_;B=NAw328B=<32M=N&0!N"6287"0=N θ Z=25©q<37>B3v7&03<1<3B37"2803>UNdZ!N (θ , θ ) <10> =87&:U6<=B37&280=>NdZ!2YC5vS2~C57&>Iv >c7 M303:"<=_8>":~4B3[\2 ] < , I 1B37&28;0=NdZ R p(u, θ)du 6:A©9B=:S4>Nª0=N&_~Nk4<=F128032;B33<30=:">N"7&`M3:~4{_;B=N&T:U6<3B37"2880!NdZ!N=, HU 1B=1:"0=6N"3<@ S<3S2;0=N:~4!B=:"8:qB!N&&KZ >L4!28B3<3@ J (θ) = R >7&:"w=TU2 θ M=:&Z=:A©q<37&28Z=`"B!Nq< p(u)du C5vS2~C57&>v"2876Nd©:"0=N"B37dN H(u) C82;628D!C57&>UN1vB3T=3<3D L(x1 , . . . , xn , θ) = ( i (n) 2;C5Z=< θ−n , 1 xi ∈ [0, θ] n (n) ∗ (n) 1 ∂2p ∂θ 2 ∂3p ∂θ 3 1 1 v84:">&Z!287&>:"0U@F> Nd@C5::"7&B=:"S2;B3 M (n) n ∂p ∂θ }4!2 1 B32F_~N&>""Z=287">":"0U@F>c7ªvUCZ=:>"<@=6 +"]=Iª<%H3, !#1'¡*^ ^ ! # ) *# )! ,..## !-,. + "+ ##' ! #%&# # # ! ' ! # ! "+' .# ! ' '&! .! ,.+ *+ ,.θ ' '& %-# ! n ! → ,∞ ( , , !-, #' !-, (θ , √ '& , /(' % ## # θ −−→ θ # e(θ ) → 1 ∗ n 1 J1 (θ0 )n P ∗ n ' , # ! , , ( % =#) 0 θ / ., '&'& %-# ! #! ) ∂L(x1 ,...,xn ,θ) ∂θ ∗ n ! # ! / N"C;C86:"7"03<=6 lim P θn∗ − θ0 p o J1 (θ0 )n 6 x = Φ(x) . @ 0=N"C8M3032425Z=2;B3<321C1MZ=:"7"B3:C87&`Q> n→∞ Γ p(u, α) = αλ λ−1 −αu u e , u > 0, α > 0 , Γ(λ) >qCZ!vUw!N&21T:"}4NªM=N&0!N"6287"0 λ <3_8>2;C57"28B^, F62;2;6 L(x1 , . . . , xn , α) = λn log α − n log Γ(λ) + (λ − 1) n X i=1 log xi − α n X xi , i=1 λ λn ∂L = − xn, α∗n = . ∂α α x α∗n @3>"Z3@3287C}@ C5:UC57&:A@37"25Z=`B3:"D^]NC5<=6M37":"7&<3w=2;C5T=< =~ @ s®: Z!N"C8B3:%4!:TN&_;N"B3B3:D7&28:032;62%M=:&Z=vw32;B3B=Nd@ :"32;B3TN 12;T37&<3>B3:"D?"[\":03TN?<3_0=NC5M303254!25Z!28B3<@CqM3Z=:"7&B=:C57"`Q> ]p_;N">"<=C}@3ª288:{:7%M=N&0=N6287&0!N ,θ 1B37&2;03>N&Z (θx ,, .θ. .), C8x:SCZ!vUw!N&D3B3[6<80=N"B3<3=N6< θ (x , . . . , x ) B=N"_8[\>UN&28p(u, 7C}@ θ)+.# ! ' # ( '&# ! ., 4=Z=@M=N"0=N"62;7&0=N θ C , + #+ '&( γ = 1 − α ]=}4!2 α ¯ )+! .#( ,"% ' ]2;CZ!< P{θ < θ < θ } = γ , ¯µB328T:7&:"0=N&@OC87AN"7&<=C87&<3TN%C91vUB=T33<32;DO0!N"C5M=0324!28Z=28B=<@ F (x, θ) = P{X 6 x} , .1vUC57&` ,...,x ) 28ª28B3<=2 xb =X(x v = & N > 3 B 8 2 B=<@ B!N&_8[\>UN&2;7"C}@ ! .,'& ( ,=.1vC87&` < ¯ !-,. ## ] 7&:2;C87&` α , α >xb 0(θ)] α + α = α ]3F<(x,M=vC5θ)7"` =xb1 −, γxb T3>N"B37&<Z!<M3:0U@4!TN α < 1 −αα , M α:"}4NªB3280!N&>"2;B=C57">"α: xzH=, I 6X 6x b x b >[\M3:&Z=B3@3287"C}@C1>2803:A@37"B3:C87&Q` > γ ]7AN"TTN"T 1 n 1 2 i 1 n 1 1 γ 1 2 n γ 1 2 1 2 α1 1−α2 1 2 2 α1 1−α2 x1−α2 , θ) = 1 − α1 − α2 = 1 − α = γ . xα1 , θ) − F (b bα1 } = F (b P{b x1−α2 6 X 6 x 1. 03254!M3:"Z=:A©q<=6¦]3w37&: 2~C57&`v[\>N?>cªN&@Ox >:"_;0=N"C87ANQ> Nd@!L1vB3T33<@%:7 ]!7&:}4N9T3>UN&B37&<3Z=` @3>&Z=@L@ 2;7"C}@>":_80=NC57dN?>cS28D¡FxzvU(x,[\θ)>NQ> S2;D!F1vB3T3=<328DM!N&0=N6287&0=N θ ,%:"B=:"7&:B3B3:C87&`µθM3: θ M3:"_;>":"Z3@3287%0=N"xb_8032;S<37"` B32;0=N&>28B=C87&>":{xzH=, I M N&T3<=6 :"0=N&_;:6¦] (θ , θ ) @3>"Z3@3θ287(xC}@,4!.:.>".28,0=x<37&, 25αZ!`")B36[6tθ 6<3B3θ7&2;(x03>N&, Z=. .:.6, x4=Z3,@{1 M=−N&0!αN"6) .287"0=N θ CGB=Nd4!25©qB3:UC57&`N> γ , N"_8"<=28B3<32 6:A©9B=:c>UN&03`<303:">UN&7&`!,ds®0=24!B@@S4=Z=<3B!N 4!:>"2;03<37&28Z=`"B3:8:c<3B37&2;03>N&Z!N M=03< =;7&:6¦]>"::"S2;:">":α0U@^, ]Uα62;B@3287"C}@^, 25ZN&7&28Z=`"B3:9C50=24!<>C528|94!:">28M03<=[θ7&25Z=−θ`B3[¦] |q<3B37&2;03>N&Z=:">C B!NA4!28©9B=:C57"`Q> γ >[\"0=N"7&` 7&:7A]&T:"7&:03[\Dª¡C50=24!FB >.> 4=Z=<3B3v,AQkZ3@9C5<=662;7&03<3w3B=[¦|ª0=NC5M303254!25Z!28B3<3D]&7&:12;C87&` 4=Z=@t0=N"C8M303254!25Z=2;B3<3D Cw32;7&B3:"D MZ=:7&B3:UC57&N` > p(u, θ) ]:"M=7&<=6NdZ!`"B3[6 0!N&_8<328B=<32;6¦]TN&Tv;4!287R>"<4!B=:RB3<3©ª2"] @3>"Z3@3287"C}@ α = α = α/2 ,YQkZ3@ χ @ 0!N"C5M=0324!28Z=28B=B3:"D C5Z=vw=N&D3B=:"D>"28Z=<3w3<3B=[E<=6A2 >c7"C}@ 7dN&&Z!<33[ >"[\:"0=N 0=N"_8<328B3<3@ α B=N"<Z=vw3ª<=6:"0=N"_8:6¦, 1 1 1 1 1 n 1 2 1 n 2 2 2 2 1 2 2 !(U#rp3j!#oqr##3#()'j#o #jt$''31!( α aa =1'j! R'3"$p j#r# .1vUC57&` x , . . . , x ¯ >"[\:"03TNk<=_FB3:"0=6N&Z=`"B=:"8:0=NC5M303254!25Z!28B3<@{C1M=N&0!N"6287"0=N"6< (m, σ) , =1'*^ ¢ , ! *% # !$#+## x !-,. #, ! *% , + # ! , # ., x !-,..! #+ # #, ! , ( , !-, #' !-, (m, σ/√n) (n−1)bs /σ ##' ! , sb!$#+ # ## (n − 1) ' ##( "+ / χ √ "+ % #' ! ,.!$#+ # # )%-, , .# % , '&( + #'&, (n − 1) ' # #( n . "+ / *! , ,' # ( ' suZ!vUw!N&D3B3[\2\>25Z=<=w3<3B3[ y = (x − m)/σ B328_~N&>"24!2~6 < ,AQS:UC57dN&7":"w3B3:GM3:TN"_;N"7&`=]Aw37&:kC5Z=vw=N&D=B=Nd@ 1 n 2 2 2 2 x−m s b i i 1 n−1 − y) >"28Z=<3w3<=B=N !< 628287c0=N"C8M303254!25Z=2;B3<32 χ (yC (n−1) C87&28M32;B3`Q> C5>:"":~4[R25Z="<=C8<=6[F,3.1vC87&` ¯:"0=7&:";:"B=N&Z=`"B=N&@6N&7&0=<3=N7dN&TNd@^]w37&: c = ] j = 1, . . . , n ] <{M3vC87&` q¯ :037&:8:"B!NdZ=`B3:"2cM30328:"0=N"_8:>N&B=<32 _~NA4N"B3B3:"21C5::"7&B=:"S2;B3<@=6< n (0, 1) y = n1 i=1 yi s21 = 2 (n−1)s1 = (n−1)b s2 /σ 2 2 y s1 C = cij (z1 , . . . , zn )t = C(y1 , . . . , yn )t n i=1 i 2 1j n M N"T{TN"T < 2 √1 n n X √ 1 X yj = ny, zi = cij yj , i = 2, . . . , n . z1 = √ n j=1 j=1 Pn j=1 cij cjk = δik ]=<=6282~6 X zi2 = n X cij M yj = 0, M [zi zk ] = j=1 Dzi = D n X n X cij yj = c2ij Dyj = yj2 j i M zi = X X cij cjk M yj2 = j=1 n X X cij cjk = 0, 2~CZ=< j=1 c2ij = 1 . M N"T3<=6t:""0!N&_8:U6¦] z , . . . , z 0=NC5M30=24!28Z=28B3[ M3:SB3:0=6NdZ!`"B3:U6v_~N&T:"B=vCcM=N&0!N"6287"0=N"6< j=1 j=1 j=1 >N"B3[F]3N_8B=NAw=<37A] z , . . . , z B32;_;N&><=C5N&@^]w37&: 1 n 1 n (n − 1)s21 = n X (yi − y)2 = i 6= k , n X z1 = √ yi2 − ny 2 = ]!CZ=vw=N"D3B=Nd@>"25Z!<3(0,w3<31)B=N ny n X zj2 − z12 = n X <B328T:030325Z!<303:?@ zj2 <=62;287µ0=NC5M30=24!28Z=28B3<=2 C C87&28M32;B3`Q> C5>":":~4![yTN&T/< , E 7CR>L4N%"[\S2C5TN&_~N&B3B3:Q@ 8: CZ=254!v"2;7~]Lw37&: z < zχ + ·(n· · −+ 1)z B32;_;N&><=C5""2;03S<37"`S4:"TN&_~N&7&28Z=`C87&>":ª7&28:032;6[ _;N6287&<=6¦]!w37&: s = CZ!vUw!N&D3B=N&@z>25/(n Z=<3w=−<3B=1)N i=1 i=1 j=1 2 2 1 n j=2 2 j 1 2 2 2 j=2 2 2 n 1 s √ √ n−1 x − m√ y n σy n Y = n= = = z 1 Pn 2 sb σs1 s1 j=2 zj @3>&Z=@3287"C}@?:"7&B3:S2;B3<32;6¡s®7&`N>L4!28B37dN9C (n − 1) C57&2;M328B=`Q> C5>:"":~4[F, .1vUC57&` B328_~N&>"[\":03TN x , . . . , x >"_8@37dN<3_RB3:"0=6N&Z=`"B=:"8:t0=N"C8M3032425Z=2;B3<@ CµM!N&0=N6287&0=N6< , (m, σ) ' !(U#rp3j !#o9n rp#3Up()'j #j $''31U#' $p#r r "()U"# K TNAw=2;C57">"2:328B=T3< >":_8`U62;6 C50=24!B32;2RN"03<=F6287"<3w32;C8T:"2 x , s®7dN&7&<=C87&<3TN x :7&>"2;w=mN&2;7>C82;6 7&032;":>N&B=<@=σ6¦]®M=0324KP^@3>&Z3@32~6[6 T :"=28B3TN"6¨UN&B37&<3Z=` B3:0=6NdZ=`B3:";:ª0!N"C5M=0324!28Z=28B=<@^]7&:q2;C87&`90328ª28B3<=2FvU0!N&>"B=28B3<@ (m, ) 1 n √σ n 1 − Φ(b uγ ) = γ . γ a& M N&T{TN&T 1 Φ(b u1−γ ) = 1 − Φ(b uγ ) = √ 2π Z∞ 2 e − u2 1 du = √ 2π −b Z uγ e− u2 2 du = Φ(−b uγ ), <=62;2;6 ub = −bu ,).1vC87&` α = α + α ,s®7dN"7&<=C57"<3TN Y = √n <=62;287B=:"0=6N&Z=`"B3:2k0=NC5M303254!25Z!28B3<32C M=N"0=N"62;7&0=N6< (0, 1) ,=.1:N=87&:U6vqB3280=N">"2;B=C57&>: −bu 6 Y 6 ub ]7&:q2~C57&` σ σ xzH=, HU m−u b √ 6x 6m+u b √ , n n >[\M3:&Z=B3@3287"C}@C1>2803:A@37"B3:C87&Q` > 1 − α , 1_qxzH=, HU®B=Nd|U:~4!<=6X4:">"2;03<37&28Z=`"B=[\D<3B=7&2803>UNdZ4=Z3@{M=N&0!N"6287"0=N m 1−γ γ 1 u bγ −∞ x̄−m σ 2 α1 α2 α2 α1 σ σ bα1 √ 6 m 6 x + u bα2 √ . x−u n n QkZ=<=B=N4!:>"280=<37&25Z!`"B3:8:q<3B37"2803>UNdZ!N0=N&>B=N (bu + ub ) ,.1:TNd©ª2~6¦]=w37": =87dNq4=Z=<3B=N"v;4!2;7B=N& α <%>[\"280=2;6 ∆ > 0 7dN&T^]w37&:"[ α − ∆ > α + ∆ 7&:"}4)N , 1_1B3280=N">"2;B=C57&> u b >u b >u b >u b σ α2 √n α1 1 α1 2 α1 +∆ 2 α2 −∆ 1 2 1 & α2 ∆ = 1−Φ(b uα2 ) − 1 − Φ(b uα2 −∆ ) = u bα2 −∆ Z 1 √ 2π e− x2 2 u bα2 ∆ = 1−Φ(b uα1 +∆ ) − 1 − Φ(b u α1 ) = 1 √ 2π C5Z=24!v287A]w37&: u bα1 Z e− u bα1 +∆ x2 2 2 u b α −∆ 1 2 dx > √ e− 2 u bα2 −∆ − u bα2 , 2π 2 u b α +∆ 1 1 dx 6 √ e− 2 u bα1 − u bα1 +∆ , 2π u b2 u b2 √ √ α2 −∆ α1 +∆ bα1 +∆ , u bα2 −∆ − u bα2 6 ∆ 2πe 2 < ∆ 2πe 2 6 u bα1 − u 7&:92~C57&` M N&T3<=6 :"0=N&_;:6¦]M303< α 6= α 4!:>"2;03<37&ub28Z=`"B3[\+D?<3ubB37"2803>UNd"2;03<37&28Z=`"B3:8:9<3B37"2803>UNdZ!Nq>:C8M3:&Z=`_8v2;6C}@:7&B3:"ª28B3<=2;6 s®7&`Qm>L428B37dN3,.1vC87&` S (t) ¯σ1vUB=T33<@%0=NC5M303254!2R@ Z=2;B3<@s®7&Q` >L4!2;B37dN{C n C57&2;M328B@=6":":~4![F, E ":"_;B=NAw3<=6w=28032;_ bt (n) T3>UN&B37&<3Z=`0!N"C5M=0324!28Z=28B=<@ S (t) ]^7": 2~C57&`9032;S28B=<32Fv0=N&>B328B3<3@ M N&TTN&T9MZ!:"7&B3:UC57&`S0=N"C8M3032425Z=2;B3<@s®7&Q` >L428B37dSNFw3(t)2;7&=B=N&1@−1vUαB=.T33<@^]7&: t (n) = −t (n) "2;03<37&28Z=`"B3:8:¦<3B37"2803>UNdZ!N¦TN&T1\M3vB3T=7&2uN^CZ=254!v"2;7\"0=N"7&` α = α = α/2 ,8.1:\7"28:"0=2;62uH=, +~®B=280=N">"28B!C57&>: α2 −∆ 1 α1 +∆ α2 α1 2 n n α n 1−γ 1 >[\M3:&Z=B3@3287"C}@C1>2803:A@37"B3:C87&`Q> C87&<3T:"D ( γ 2 sb sb −√ b tα/2 (n − 1) tα/2 (n − 1) 6 x − m 6 √ b n n 1−α , E 7"CA>L4NB=Nd|U:~4!<=6O4:">"2;03<37&28Z=`"B=[\D?<=B37&280=>NdZ4=Z3@{M=N"0=N"62;7&0=N sb sb x− √ b tα/2 (n − 1) 6 m 6 x + √ b tα/2 (n − 1) n n !(U#rp3j!#o9ntr#p3U#('j j#$''31U#' σ $##r r " ()&! X xk − m 2 s20 = , σ2 σ n n k=1 m m KL:C5M=:&Z=`_8v"2~6C}@C57dN&7& S2;D χ @ 0=N"C8M3032425Z=2;B3<32cC n C87&28M32;B@=6<C8>":":~4![F, E :"_;B=NAw3<=6Ow32;0328_ K (x) 1vB3T3=¨0!N"C5M=0324!28Z=28B=<@ CZ!vUw!N&D3B3:D>25Z=<=w3<3B3[ s /σ <{w32;0328_ bk (n) ¯T=>N&B=7&<Z=` K (x) ]!7&:q2;C87&`9TU:0328B=`v0=N">"B32;B3<@ 2 n 2 .1vC87&` α = α1 + α2 2 α n Kn (x) = 1 − α . , M :"}4NB32;0=N&>28B=C87&>": 2 >"[\M=:&Z=B@32;7"C}@SC®>"280=:A@37&B3:C87&`Q> <3B37"2803>UNdZ?4=Z3@?M=N&0=N6287&0!N σ 1−α s b k1−α1 (n) 6 n 02 6 b kα2 (n) σ ,~N"_8032;ªNd@!=;7&:GB3280=N">"2;B=C57&>: :7&B3:C8<37&28Z=`"B3: σ B=N&|:~4!2803<37"25Z=`B3[\D s ns20 6σ6 s ns20 %:A©qB3:4!:TN"_;N"7&`=])w37&: =;7&:"7{<3B=7&2803>UNdZ/<=62;287{B=N& C503254!BF>.>y4=Z=<3B3vOC503254!C825|%4!:>"2;03<37&28Z=`"B3[¦| <3B37"2803>UNdZ=:> =;7&:";:F>"<4)NFC®4!:">2803<37"25Z=`B3:"D9>"2;03:A@37&B3:UC57&`N> 1 − α ]"2;CZ!< α < α >[\"0=N"B3[t7dN&T]"w37&:FM3Z=:"7&B=:C57"` v;4!:>&Z=2;7&>":0@32;7ªC8:":"7"B3:"ª28B3F< > k (x) = K (x) b k1−α1 (n) 1 n n b kα2 (n) 2 kα2 (n)). k1−α1 (n)) = kn (b k n (b ! (U#rp3j!#o9n¡rp#3Up()'j j $''31U#' σ $p#r Ur (& m K®:UC5M3:"Z=`"_;v"2~6C}@7&28:032;6:D H3,-+;=]^C5: Z!N"C8B3:T:7&:"03:D/C57dN&7& C5>:"":~4[F, ¬FN&T?<{>9M3vB3T37&2F>(=;7&:qM303<=>":~4!<37T4!:">(n2803<=−7&251)Z=`B3:6v<3χB37&2;03>N&Z=v s b2 σ2 Cc4!:>"280=<37&25Z!`"B3:D>"2;03:A@37&B3:UC57&`N> sb s 2 n−1 b kα2 (n − 1) 6 σ 6 sb s n−1 b k1−α1 (n − 1) x , r#n $ ! ¨ Y=()Upr#3j !Glt!²#U#rp()'#r .103<¡:&Z=`S:U6:"NP^2;62%>[\":03T3< :"32;B3T3< 6287":~4N/6:62;B37&:">t<¡B=N&<3:&Z=`S2;8:OM30=N&>&4!:"M=:~4!:"<@^]TN&T M30=N">"<Z!:=]pNC5<=6M=7&:"7&<=w32;C8T3<B3:0=6NdZ=`B3[F, E C87dN&B3:>"<=6C}@XM3:~4!03:"B3:{B=N0=N"C;C86:7&0328B=<3&4!:"M3:~4:""<3@^,~K/vUCZ=:>"<@|k7"28:"0=2;6[H3,-+~V :"32;B3TN θ N"C5<=C5<37:"7 θ <0=N">"B=N σ ]"v;4!2~6 C5w3<37dN"7&`=]w37&: θ B3:"0!6NdZ=`B3: 0=NC5M303254!25Z!28B3B3:D?C1M!N&0=N6287&0=N6< (θ, σ/√n) ,=.10=< =;7&:64!:"M=vUª28B3<=<4!:>"2;03<37&28Z=`"B3[\D?<3B37"2803>UNdZ%<=6282;79>"<4 2 1 J1 (θ) 2 ∗ n σ σ θn∗ − u bα/2 √ 6 θ 6 θn∗ + u bα/2 √ n n C\M30=<3&Z=<=_8<37&28Z=`"B=:"DqB=Nd4!25©qB3:UC57&`N> γ = 1 − α ]&}4!2 ub ¯ªT3>UN&B37&<3Z=`kM3:0@4TN α B3:0=6NdZ=`B3:";:10=N"C8M3032425Z=2;B3<@ C1M=N"0=N"62;7&0=N"6< (0, 1) , OuC5Z=< σ (θ) _;N">"<=C8<37:"7 θ ]3v842;6 1vB3T33 η = g(θ) 7dN&T=vI>ª]3w37&: ] +A E 328B3TN η = g(θ ) M=N&0=N6287&0!N g(θ) N"C8<=6M37&:7&<3w32~C5T3<B3:"0=6N&Z=`"B!NqCFM=N&0!N"6287"0=N"6< g(θ), M303<3w32~6 g (θ)σ(θ) B321_~N&>"2;C57"B3:";:M!N&0=N6287&0=N θ , I kvB3T=3<@ g C57&03:8:q6:"B=:"7&:B3B=NB=N96B3:A©ª2~C57&>2c4!:"M3vUC57&N3]Uw37&: g 0 (θ)σ(θ) = 1 ]7&:q2~C57&` g(θ) = g(θn∗ ) Z dθ . σ(θ) N"C5"2;B=C57&>: 1 1 ηn∗ − u bα/2 √ 6 g(θ) 6 ηn∗ + u bα/2 √ n n >[\M3:&Z=B3@3287"C}@C1M=03<3&Z!<3_8<37"25Z=`B3:"D?B=NA425©qB3:C87&`Q> γ =1−α ,=K 0=28_8v~Z=`d7dN"7&2"]<3B=7&2803>UNdZ 1 1 6 θ 6 g −1 ηn∗ + u g −1 ηn∗ − u bα/2 √ bα/2 √ n n B=N"T303:287M=N&0=N6287&0 θ CGM303<3"Z=<3_;<37&25Z!`"B3:D?>2803:A@37"B3:C87&`Q> γ , ! # ! QS:">2803<=7&25Z=`B3[\D<=B37&280=>NdZ?4=Z=@>"2;03:A@37&B3:UC57&C}|U2;62 SL2803B3v~Z3Z!<^, .1vUC57&` µ ¯%w3<=C5Z=:{/ vC5M=25|:>C9C52;03<3< n B32;_;N">"<=C8<=6[¦|µ<=C5M=[\7AN"B3<3DXCB328<3_;>"2~C57&B3:DR>2803:A@37"B3:C87&`Q> p vUC5M328|UN> :~4!B=:6<=C5M3[\7dN"B3<3<^, E 328B3TN p = M!N&0=N6287&0=N p M3:&Z=vw32;B=N62;7&:~4!:6?B=N"<3":"Z=`"ª28;:®M30!N&>d4:"M3:~4!:"<@^, E =28B3TN N"C5"0=Nd¯ kN"MZ!N"C;N%C?M=N&0=N6287&0!N"6< p, q ,YKL[\"2;032;6 p 1vB3T3=F< > g ]30!N&>"B=Iv > Z ∗ µ n p(1−p) n ∗ M :"}4N:"=28B3TN g(p) = 2 arcsin pµ n p dp √ = 2 arcsin p . p(1 − p) NC5<=6M=7&:"7&<=w32;C8T3<%B3:"0!6NdZ=`B=NªC1M!N&0=N6287&0=N6< 2 arcsin r bα/2 µ u √ − √ 6 2 arcsin p 6 2 arcsin n n >[\M3:&Z=B3@3287"C}@µCªM303<3"Z=<3_8<=7&25Z=`B3:"D>"280=:A@37&B3:C87&`Q> 7&28Z=`"B=[\D<3B=7&2803>UNdZ?4=Z3@{M=N"0=N"62;7&0=N p γ = 1−α bα/2 µ u + √ n n √ √ u bα/2 u bα/2 sin2 arcsin p∗ − √ 6 p 6 sin2 arcsin p∗ + √ . 2 n 2 n ,=b1280!N&>"2;B=C57">": , E 7"CR>L4)NM3:"Z=vw=N&2~6M303<3"Z=<©ª2;B3B3[\Dµ4!:">2803 _;Nd4N&B3:t_;N&0!N&B32;2", B=28:"A|U:~4!<=6: M30=:">"2;03<37&` C57dN&7"<=C57&<=w32;C8T3v > ;<3M3:7&28_;v"]¦w37&: B=N&N>[\":"0=TNM3:&Z=vw32;B=N <3_R;28B32;0=NdZ!`"B3:D C5:>":"T=vUM=B3:C87&< < =;7&<=6 0=NC5M30=24!28Z=28B3<=2;6¦, OuC5Z="2;03B=N=]p7&:=;6M=<303<3w32~C5TNd@ 1vUB=T33<@0=NC5M30=24!28Z=28B3<3@ F (x) @3>&Z3@L@ 2;7"C}@OM=03<3&Z!<©ª2;B3<32;6¨>%TN&T:6 @ 7&:%C86[C5Z=2T/1vB3T33<=< F (x) M=03< n → ∞ ,.10324M3:&Z=:A©q<=6¦]w37&:µM3:UC57&0=:"28B!N 62;0=N:"7&TZ!:"B32;B3<@ :"7 ¯C57dN&7&&Z=@3287"C}@{1vUB=T33<32;D:7>[\":03T3<<{M30=24=Z=<=>":C87&"B=F:"DO(x)8<3M3:7&F28_;(x)[0!N"C5M=0324!28Z=28B=<372 S=87&:DC57dN&7"<=C57&<=T32;C87&B3:%7&:w3B3:%<Z=<NC5<=6M=7&:"7&<=w32;C8T3< 7&:w3B3:!,KL[\"2;032;6 w3<=CZ!: S 7dN&T^]w37&:"[ P{S > S } = α , OuCZ!< α 6NdZ!:=]7&:?6:A©2;6 C8w3<37dN&7"` !-,"! ' %-# ! +'&.# ! ,"! '& ]&w37&:kC8:"[\7&<32\C¦>"2;03:A@37&B3:UC57&N` > B32\M303:<3_8:D4!2871>F254!<3B3<3w=B3:6R:"M3[\7"2",".1:1>"[\:"03:Lw @ B3[6 _8B=NAw32;B3<@=6>[\w3<=CZ!<=6¡>"28Z=<3w3<=B3v S H, OuCZ= ª<|UC}@?4N"B3B3[¦|,!K¨M303:7&<3>B3:6CZ!vUw!N&2k;<3M3:"7"28_8v?:"7">"280=5N&2;6¦, E M3<=C8N"B3B3[\D%T30=<37&280=<3D B=N"_8[\>UN&287C}@ !+! ' # ! # .,"% '&* ") ,. ,d.FN&0=N6287&0 α B=N&_;[\>N&2;7"C}@ªv03:>"B32~6µ_;B=NAw3<=6:UC57&<^,.1:~C4 @ w32;03T3B32~6t:UC5:":=]Uw37&:qM=03:">2803TNª8<3M3:7&28_;[¨B=2F2;C57"`S4:"TN&_~N&7&28Z=`C87&>":ª8<3M3:7&28_;[F, E 4B3<=6 <3_1B=N&<3:&Z=2;21<3_8>2;C57"B3[¦|T303<=7&2803<=28>C5: Z!N"C8<@@3>"Z3@3287"C}@ χ @ T=03<37&2;03<3D%.1<30=C8:"B=N=, {#rp3U#rpn (&j#lm'3 χ ∗ n ∗ n 2 2 =+ $3j#& =$## YjU## r#$#!3Urpm&^ X#'=&$## Y"jUpr# #pr#3Upr#n r##=' χ :"0=N"_8v > 79>[\":"0=T3v<3_%_8B=N?@ N"C;C86:7&03<=6;<3M3:"7"28_8v ]=C8:C87&:A@3Sv > >7&:U6¦]!w=7&:94N"B3B3[\2 w32;B3<3D/CZ=vw=N"D3B3:DR>25Z=<3w=<3B3[HCS4N"B3B3:DO1vB3T3=<328DX0=NC5M303254!25Z!28B3<@x F, . .=. , Fx , E M30=24!28Z=<=6 62803vµ0!N"C}|:A©4!2A@ B3<@/7&2;:"032;7&<3w32~C5T:";:µ":%_;B=NAw328B=<3D/CZ=vw=N"D3B3:"D/>25Z=<=w3<3B3[ ξ 0=N"_8:"`"2~6¡B=N r M=03:628©9v7&T:> I , I , . . . , I ,.1:{<3_8>2;C87&B3:"D/1vB3T3=<3< F >"[\w=<=CZ=2803:A@37&B=:C57":S>"[\:"03:w3B3[¦|9_8B=NAw=28B3<3D]UM=:"M=Nd4?N >cS<|>ªM303:628©qvU7":"T , P{ξ ∈ I } = 11C5B=:=]w37&: Pp ν = nν, E 7&B3:C8<37&28Z=`"B=N&@?w!N"C57":"7dNM3:"M=Nd4N&B=<@?_;B=NAw32;B3<@>[\":03T3<{>9M303:628©qvU7":"T I 0=N&>IB=N ,=N96280=v0=NC}|:A©4!28B=<@625©4!v>"[\:"03:w3B3[6/0=NC5M30=24!28Z=28B3<=2;6 "B=[6 n/pi i χ b2n = , M :"}4)NªC87AN"7&<=C87&<3TN3]B=NS:C5B=:">"2 TU:7&:"0=:"DC87&03:"<=7"C}@T303<=7&2803<=D r X (νi − npi )2 npi r X ν2 −n, npi i=1 = M303:UC57&:9>[\0=Nd©9N&287C}@w3280=28_1=;6M=<303<3w32~C5T3<=2Fw=N"C87&:"7&[ ν {< 8<=M3:"7&2;7&<3w32~C5T3<=2Fw=N"C87&:"7&[ =1'*^ ^ ! !-,..# + '& ) ' # . H ! ,.!$#+ # ## χb ! ! !-,..! # +# # (r − 1) ' χ 2 *! , ,' # ( ' O®CZ=<{8<=M3:"7&2;_;N 0=NC5M303254!25Z!28B3<32 / i=1 ##( "+ / H npi i 2 n >"2;03B=N3]7&:C5Z=vw=N&D=B3[\21>"25Z!<3w3<3B3[ P{νi = ni , i = 1, . . . , r} = ν1 , . . . , νr , n→∞ , "+' <=62A>c7M=:&Z=<3B=:6<=N&Z=`"B3:2 n! pn1 . . . pnr r , n1 + · · · + nr = n . n1 ! . . . n r ! 1 1_17&28:032;6[ I, H"CZ!24!v287A]3w37&:C5Z=vw=N&D=B3[\D>28T37":"0 (n) ηk (n) (n) η (n) = (η1 , . . . , ηr ) = GT:6M3:B328B37dN6< νk − npk √ npk T{B3:0=6NdZ=`B3:6vq0!N"C5M=0324!28Z=28B=C1B3v~Z=28>[6UN&03c7ªT:U6M3:"B32;B37&[ CZ!vUw!N&D3B3[\2F>25Z=<=w3<3B3[°CFM303254!25Z!`"B3[6 B3:0=6NdZ=`B3[6 0=N"C8M3032425Z=2;B3<32;6¦, cN"0=N&T=7&η2803"B=N C}|:~4<37"C}@M30=< n→∞ k ` k` k ` k 1 e− 2 Bη (t) , }4!2 Bη (t) = r X t2k − .1vC87&` C ¯:037&:";:"B=N&Z=`"B!Nd@6N&7"03<3=N9><4N k=1 X k,` 2 X√ X √ tk t` p k p ` = p k tk . t2k − √ p √ 1 p2 .... √ pr c21 c22 ... c2r r r k=1 k=1 ··· ··· .... ··· cr1 cr2 ... crr UI 1. :"Z=:A©q<=6 < ,YGN&0=N"T37&2R@ . , u ) = u = t C = (t , . . . , t ) C 03&=@3_;N"(ηB3[ , .C5.:. :", 7&ηB=):"CS2;B3<3(u2~6 ,f. .(u) Y , 1 . N : 8 = & 7 U : 6 v ] = f (u C ) f (u) = e }4!2 1 r 1 ξ r 1 r η ξ Bξ (u) = Bη (u C 0 ) = r X t2k − u21 = r X 1 η u2k − u21 = r −Rξ (u)/2 ξ r X u2k , 7A, 2", Dξ = 0 25Z=<=w3<3B3[ ξ , . . . , ξ 0=NC5M30=24!28Z=28B3[ B3:"0=6N&Z=`"B=:CM=N&0!N"6287"0=N"6< (0, 1) <=C5<=6[F, 1_GM303254!25Z=`B3:"D7&28:032;6[¨7&28:03<3<62;03[ <<3B37&2;80=N&Z!N?xz1:"0!6v;Z!<303:">TNkT:"7&:03:"DM303<3>24!2;B=NkB=<©ª2A C5Z=24!v287A]w37&:?M3:C5Z=24!:>N"7&25Z=`B3:C87&`90=N"C8M303254!25Z=2;B3<3DOCZ=vw=N"D3B3[¦|%>"28Z=<3w3<3B C}|U:~4!<37"C}@%T 0=NC5M30=24!28Z=28B3C< >yC5Z=vw=N&D3B=:"Dµ>25Z=<3w=<3B3[ η + · · · + η ,K C8<Z=v%:"0=7&:";:"B=N&Z=`"χbB3:UC5=7&<µη 6N"+7&03·<3·=· [ + Cη <=62;287{62;C87&: 0=N">"2;B=C57&>: η + · · · + η = ξ + · · · + ξ = ξ + · · · + ξ ]vUw=<37&[\>N&@^]w37&: ξ = 0 M^, B^,W 7&>280U©4!28B3<=2c7&2;:"032~6[ C5Z=24!v2877&28M=2803`q<3_F:M303254!25Z=2;B3<@ χ @®0=N"C8M3032425Z=2;B3<@^, bFNªM30=N"T37&<3T2F_~NA4QN > 7"C}@v03:>"B32~6_;B=NAw3<=6:UC57&< ,)s®:" Z!NC5B3:7&28:032;62FH=, +daFM30="280=:A@37 @ B3:UC57&N` >S]M30=<3&Z=<3©ª28B=B3:90=N&>B3:"D α ]=>"[\M3:"Z=B@3287C}@B32;0=N&α>28B=C87&>": k=1 1 2 2 1 2 r 2 1 2 2 r k=1 k=2 r 2 1 χ b2n = 2 2 2 r 2 n 2 r 2 n1 2 nr 1 r X (νk − npk )2 >b kα (r − 1), npk }4!2 k (r − 1) ¯ α@ T3>N"B37&<Z=` χ @ 0=N"C8M303254!25Z=2;B3<@C (r − 1)@ DOC57&2;M328B3`N>yC8>":":~4![F,FO®CZ="280=B=N3] 7&:%M=0=N&T37"<3w32;C8T3:"_~6:A©9B=:?>UC57&0=287&<37"`%>%24!<3B=<3w3B3:DO>[\":03T2_;B=NAw32;B3<32 χb ]^M=0328>[\ªN?>cS2;2 k (r − 1) , .1N: =87&:U6vt:C8B3:">B=Nd@ 8<3M=:"7&2;_;NC8w3<37dN&2;7"C}@ M=03<3B@37&:D <Z=<¡>[\":03TNOC8:">U62;C57"<=6NC%;<3M3:"7"28_8:D^]u2~CZ=< χb < <{:"7&>2803;B3v7&:"D^]2~CZ=<{>[\M3:&Z=B3@3287"C}@?M303:"7"<3>":M3:&Z=:A©qB3:2 B32;0=N&>28B=C87&>":!, k (r − 1) KL[\:"07&:"w=28T z , . . . , z B=NM30=N&T=7&<3T2:UC5vS2~C57&>"Z3@3287"C}@7dN&T3"Z=287">":"0=<37&` B32;0=N&>28B=C87&>N6 2 α k=1 2 n α 2 n α 1 r−1 ! # ! K/M3:UCZ=254!:">UN&7&28Z=`"B=:C57"< B32;_;N">"<=C8<=6[¦|S<=C8M3[\7dN&B3<3D9C5:"[\7&<=2 M303:<3_8:kZ=: ν 0=N&_,s®:"Q> @ 62~C57&<=6:9Z!<3=;7&:qC1/ 8<3M=:"7&2;_8:"D]w37&:C5:"[\7&<=2 nA °>"280=:A@37&B3:C87&` A ¬FN&©4!v >ETU:U6M3:"B=28B37&v >"[\:"03T3<v842;6y0=N"C;C86N"7&03<3>UN&7&`/TN&T<3B4!<3TN&7&:0¡C5:"[\7&<@ , <3M=:"7&2;_;N 1:0=6v~Z=<303v287"C}@CZ!24!v > ª<=6°:"0=N&_;:-6 J^>2803:A@37"B3:C87&"B3[ p < q C5::"7&>A287"C87&>"2;B3B3:!,p.1H:?@ C87&03:" 10, νk > 10 . χ b2n = (ν − np)2 (n − ν − nq)2 ν2 (n − ν)2 (ν − np)2 + = + −n= . np nq np nq npq Nd4NA4<=6C}@Sv03:">B32;6_8B=NAw=<=6:C87&< <9M3:F7dN&&Z!<332¦_8B=NAw=28B3<3D @ 0!N"C5M=0324!28Z=28B=<@9CL:~4!B=:"D9C57&2;M328B3`N> C5>:"":~4[ B=N"D4!2~6°T3>N"B37&<Z=` k (1) ,s®0=N">"B3N"B37&<Z!`Q> k (1) , OuC5Z=< χb < k (1) ]p7": v842;6tC8w3<37dN&7"`q>"[\":03T3vC8:">62~C57&9M30=:"7&<3>B3:6 CZ!vUw!N&21:"7&>2803;B32;6t282, .1vUC57&` F (x) ] x = (x , . . . , x ) ¯ 1vB3T33<@¡0=NC5M30=24!28Z=28B3<3@ CZ=vw=N"D3B3:";:/>"2;T37&:"0!N ξ = (ξ , . . . < ¯{1vUB=T33<@%0=NC5M303254!25Z!28B3<@C5Z=vw=N&D=B3:";:q>"28T=7&:"0=N ξ = (ξ , . . . , ξ ) ,.1:UCZ=254!:">UN&7&28Z=`"B3:UC57&` . . . , ξ ) F (x) C ! Z & N O : } C U | ~ : ! 4 <37"C}@ T x ]Y2~CZ=< F (x) → F (x) >XTNd©4!:Dt7&:w3T2?B=28M3032;03[\>"B=:C57"< F (x) M30=24!28Z=`"B3:D?1vB3T=3<3<^, F (x) F (x) ⇒ F (x) !#1'?* h!´ x c25Z3Z!"2;03:A@37&B3:UC57&B=[\2u62803[F]~M3:03:A©4!28B3B=[\2u1vB3T33<@=6"<=C8<37/:"7 M=N&0=N6287&0=:">=, <=M3:"7&2;_;N H ]=M3:~4=Z=28©N&Nd@M=03:">2803T2"]3C8:C87&:"<379>q7&:U6¦]3w37&:>"[\"F:03(x,TN θx , ,. .. .. ., ,θx) <=_8>&Z!28w32;B=N<3s_F82;B3280=N&Z=`"B=:"D C5:>":T3vM3B3:C87&<CF1vB3T=3<328D?0=NC5M303254!25Z!28B3<@ F M=03<{B328T:"7":"03[¦|_;B=NAw32;B3<@|M=N"0=N"62;7&0=N=, ¬FN"TFM303254![L4!vS2~6?M3vB3T37&2®"v;4!2;6%C8w3<37dN&7"`=]8w=7&: >[\":"0=TNL0=N"_8<37dN\B=N 80=vUM=M^]~C8:":7&>"2;7"C57&>v > S<3| B32;M328032~C52;TNQ> ª<=6C}@GM=03:628©9v7&TN6 I , . . . , I ]~.1vUC57&` ν T:&Z=<=w32;C87&>":\>[\":"0=:"w3B3[¦r |c_8B=NAw32;B3<3D^];M3:"M!N&>"ª<|c> I r, .103< θ , . . . , θ "280=:A@37&B3:C87&"28Z=<3w3<3B=[ > M303:U625©qv7&:"T ,¦.FN&0=N6287&0=[ B32;<3_8>2;C57"B3[E<4!:"Z3©qB3[E"[\7&` :"32;B328B=[ M=: >"[\:"03T2,Yb1:/2;CZ!< θ _;xN"62;B3<37&`/<3| :328B3TNI6<^]Y7&:/7&:}4N pθ C57dN&B3v7C5Z=vw=N&D3B=[6<¡>"25Z!<3w3<3B=N6<< 6[ B=2 C86:A©2;6yM303N"287"C}@^]2~CZ=<¡>TNAw32~C57&>2%:"32;B3:"TM=N"0=N"62;7&03:> >"_8@37&`:328B3T=< 6N&TC5&4!:"M=:~4!:"<@ θ ]®7&: TN&T¡M3:TN"_;N&Z #, S<3S2;0 C57dN"7&<=C57"<3TN χb M303254!25Z!2FM303< n → ∞ 0!N"C5M=0324!28Z=28B=<32 χ C r − s − 1 C57"28M32;B@=6<µC5>:"":~4[F, =1'R* hh xz#, S<3S2;0!, ' ,' '& ! , χ2 1 s 1 1 r i 1 s k i i 1 s i ∗ in χ2 2 r ∗ ∗ X νi − npi (θ1n , . . . , θsn ) = ∗ , . . . , θ∗ ) npi (θ1n sn i=1 χ b2n r−s−1 i 2 n '&## . "+ / {pr#3Upr#n %=j1 =!(' QS03v;<=6tB=28M=N"0=N"62;7&03<3w32~C5T3&Z=@3287"C}@ !+! '&# ! £®7&:79T303<37&2;03<3D%:UC5B3:>N&BB=N9C57dN&7"<=C57&<=TU2 Dn = Fn∗ (x) i i 2 }4!2 k i i ##' ! , !$#+ # ## n sup ¯7=~6M3<303<3w=2;C5TN&@{1vB3T3=<@0=NC5M30=24!28Z=28B3<3@ −∞c":"D9B32;M3032;03[\>"B3:D <=62;28717dN&T:"2L©2\0=N"C8M3032R@ 4!28Z=28B3<=2"]UTN"T<>CZ=vw=N"2"]UTU:}4N F (x) :M303254!25Z3@3D2;7k0=N">"B3:U62803B3:20=NC5M303254!25Z!28B3F<3(x) 2cB=N [0, 1] ,.1vC87&` x , . . . , x ¯µ>"[\:"03TNC0=NC5M30=24!28Z=28B3<=2;6 F (x) ,p.10324!M=:&Z=:A©q<=6¦])w37&: F (a) = 0 ] F (b) = 1 < F (x) C57"03:";:?>:"_;0=N"C87AN"287 B=N (a, b) ]!M30=<3w32;6 a "254!2;6¡C5Z=vw=N&D=B3[\2S>25Z=<3w=<3B3[ y = F (x ) , M N&TRTN&T B328_~N&><=C5<=6[F]=7&:9< B328_~N&>"". ,B3y:U62803B3:90=NC5M30=24!28Z=28B3[¨B=N [0, 1] ,3.103<CZ > :6 y ∈ (0, 1) <=6282~6 n 1 k 1 n 1 1 QkZ3@ n n P{yk 6 y} = P{xk 6 F −1 (y)} = F (F −1 (y)) = y y = F (x) .1:Q=;7&:6v CZ!vUw!N&D3B3[\2FC8:""[\7"<@ {yk 6 y} Fn∗ (x, x1 , . . . xn ) = < {xk 6 x} 0=N">"B3:UC5<Z=`B3[F,.1:Q=;7&:6v n n k=1 k=1 1X 1X I{xk 6x} = I{yk 6y} = Fn∗ (y, y1 , . . . yn ) . n n Dn = sup |Fn∗ (x, x1 , . . . xn ) − F (x)| = sup |Fn∗ (y, y1 , . . . , yn ) − y| . a9TNAw32;C87&>"2 F (y) CZ=254!v"2;79>"_5@37"` ¡ ¥S, b9, ¬1:&Z6:"8:03:">k4!:"TN"_;NdZp]Uw=7&: −1 sup{x : F (x) = y} , ∞ X √ 2 2 lim P{ nDn 6 x} = K(x) = (−1)k e−2k x , x > 0. n→∞ Fb Nt:C5B=:">"2 =87&:8: M3032425Z=`B3:";:t0=N">"28B!C57&>UNtC57"03:"<37C}@ T303<37"2803<3D°¬1:&Z6:"8:03:">UN3,L.1vC57"` k ¯ α@ T3>UN&B37&<3Z=` M30=24!28Z=`"B3:8:0=N"C8M303254!25Z=2;B3<@ ) = α. M :"}4N8<3M=:"7&2;_;N:?7":6¦]w37&:{>[\":"0=TN>"_5@37dN<31_9−0=K(k N"C8M303254!25Z=2;B3<@ F (x) ]M303<3B3"2;035N&2;7"C}@^]2;CZ!< √nD > k ,Wc03:">28B3`9_8B!NAw3<=6:C87&3< =87&:8:9T303<37"2803<@%M30=<3&Z=<=_8<37&28Z=`"B=:0!N&>"2;B α , {pr#3Upr#n iªcr###!(' .1vUC57&` < ¯4!>2SB32;_;N&><=C5"[\:"03T=<^,!.12;03>N&@{7&:"0=N&@>"[\:"03xTNk, .¯ . .G(x) ,!KLy>"25,4!. 2;. 6. , yC57dN&7&4!:TN"_;N&Z]w37&:O2;CZ!< ]=C5Z=vw=N&D3B!Nd@>25Z=<=w3<3B=N 0<τ <∞ sup −∞B3[F]7&:XM303< n, m → ∞ 7AN"T^]w37&: n m → τ nm Dn,m n+m ] C87&032;6<=7"C}@ T 0=NC5M303254!25Z!28B3 ,L£®7dNOM3032425Z=`B=Nd@ 7&28:032;6NOM3:_8>":"Z3@3287OC57&0=:"<37&`/T=03<37&2;03<3DM303:>"280=T3< ;<3M3:"7"28_8[F]w=7&:q>"[\:"03T3< x , . . .K(x) < >_5@37&[¨<3_F:~4!B3:8:<7":"8:9©210=N"C8M3032425Z=2;B3<@^, ,x y ,...,y {#r#3#r#n #'m#r#1&r j# $'#'=GU=( +A.1vC87&` < >"[\":03T3<9<3_\B3:0=6NdZ!`"B3:D9C8:">:"T3vM3B3:UC57&<9C¦B32;<3_8>2;C57"B3[6XC503254!B3<=6<^, QkZ3@?M303:>"280=T3<;x<3M3,:. 7&. 28. _;, [x H :yσ, . .=. ,σy <=C8M3:&Z!`"_8v287"C}@?C57dN&7&:"":~4[F,.1vC87&` T3>UN&B37"<Z=`90=N"C8M303254!25Z=2;B3<@ k<=S280!N3]37&:92~C57&(n`q03−2;S1,2;mB3<3−21v1)0=N&>B328B3<3@ N6287&UN&B37&<3Z=28D F62;2;6 P{Fn,m > Fbα (n, m)} = α . Fbα (n, m) Fbα (n, m) :":"_;B=NAw3<=6 >[\M3:&Z!B@3287"C}@?C5:":7&B3:S28B=<32 Fbα (n, m) = 1 . Fb1−α (m, n) P{Fb1−α/2 (n, m) < Fn,m 6 Fbα/2 (n, m)} = P{Fn,m 6 Fbα/2 (n, m)} − P{Fb1−α/2 (n, m) < Fn,m } .1:"Z=:A©q<=6 =1 − α/2 − (1 − (1 − α/2)) = 1 − α/2 − α/2 = 1 − α . C1 = Fb1−α/2 (n, m) < C2 = Fbα/2 (n, m). UV OuC5Z=<4=Z3@{M3:&Z=vw32;B3B3[¦|>[\":03:"T>"[\M3:"Z=B@3287C}@B32;0=N&>28B=C87&>": C1 6 sb2x 6 C2 sb2y 7&:q;<3M3:"7"28_;NM30=<3B3<=6N"287"C}@^]3>9M30=:"7&<3>B3:6 CZ=vw=N"218<3M3:7&28_~N:"7&TZ!:"B@32;7"C}@^, IQkZ3@¡M303:>"2;03T3< ;<3M3:"7"28_8[ : 0=N&>28B=C87&>"2{4<=C5M32;0=C5<=D x H : σ = σ 4!Z3@ >"[\:"03:T x , . . . , x < <3_FB3:0=6NdZ=`B3:"D?C5:>":"T=vUM=B3:C87&<C1<3_;>"2~C57&B3[6<%C80324!B=<=6<{<=C5M3:"Z=`"_;v"2;7"C}@C57dN"7&<=C57"<3TN y ,...,y 1 2 x 2 y 1 n m s20x , s20y Fn,m = ¬1:"7":"0=N&@<=6282;7q0=N"C8M303254!25Z=2;B3<32 S<3S2;0=N9C (n, m) C57"28M32;B@=6<%C5>":":~4![F, HU <3M=:"7&2;_;N1:k0=N">"2;B=C57&>2C503254!B3<|%x H : m = m M30=<<3_8>2;C87&B3[¦|k4<=C5M32;0=C5<3@| C1<=C8M3:&Z=`_8:>N&B=<32;6/C87AN"7&<=C87&<3T3< x Z=p y x−y D(x − y) =q x−y 2 σx n σy2 m σx2 < σy2 M30=:">"2;0U@3287"C}@ . ®s 7dN&7&M303:7&<3>"B=:6 C5Z=vw=N&2,)!42;C5` ub ¯{T3>N"B37&<Z!`B3:0=6NdZ=`B3:";:0=NC5M303254!25Z!28B3<@OC |Z| "2;B=C57&>2C80324B3<|µM3032;C57"B3:"Dµ4!<=C5M=280=C8<3?M303254!M3:"Z=:A©ª2;B3<3< 0=N">"28B!C57&>UNF4!:"":~4[F, M :"}4NqC87AN"7&<=C87&<3TN α/2 + α/2 2 x 2 y 2 2 x 2 x n i=1 2 y 2 y 2 i x 2 i y 2 (n − 1) sb2y sb2x + (m − 1) σx2 σy2 0=NC5M303254!25Z!28B=NM3:9_;N"T:"B3v χ C (n + m − 2) C57"28M32;B@=6<µC5>:"":~4[F, N6287&&Z=@3287"C}@?B32;C;628S2;B3B3:D?:328B=TU:Dq4!25Z=<3w=<3B3[ 0=N">"B3: 4!:M3vS28B=<@|0=N">"B=N σ + ,=.1:Q=;7&:6vCZ=vw=N"D3B=Nd@?>"28Z=<3xw3<3−B!yN 2 1 n 1 m m x − my <4!"28Z=<3w3<3B!N √ n+m−2 B32¦_;N">"<=C8<37G:"7 m ] m < σ <0=N"C8M303254!25Z=2;B=NGM3:ks®7&`Q>L4!2;B37&vC 8<=M3:"7&2;_8[ H : m = m ":":~4![F,8QkZ3@M303:">2803T=< y τn+m−2 = q (x − y) (n − 1)b s2x + (m − 1)b s2y √ n+m−2 r mn , n+m 0=NC5M30=24!28Z=28B3B!Nd@M=:s®7"`Q>L4!2;B37&vC C57&2;M328B3@=6<{C8>":":~4![F, ²#^'R^ - - rp²#r#U#'^=##j²pr#r .1vUC57&` (x , y ), . . . , (x , y ) ¯B=28_;N">"<=C8<=6[\2OB=N&"CZ >L4!28B=<@ B=NA4¨4!>"v62;03B3[6 C5Z=vw=N&D=B3[6 >"2;T37&:03:6 ,=KL>"242;6tC87dN&7&<=C87&<3T3v (X, Y ) 1 1 n (n + m − 2) n r∗ = }4!2 m11 , sx sy n m11 = < n 1X 1X (xi − x)(yi − y), s2x = (xi − x)2 n i=1 n i=1 ¬FN"TM3:TN"_;N&Z S<3ª280{C57dN&7&"28Z=<3w3<3B 2 X < Y ,=bFN&M=:6B3<=6¦]w=7&:1vUB=T33<@ 1+r 1 log 2 1−r @3>"Z3@3287"C}@%:"0=N"7&B3:"D{Tµ;<3M32;03":"Z=<3w32~C5T:6v?7dN&B3;28B=C8v",CO®CZ=< xb ¯?T3>N"B37&<Z=`B3:"0=6N&Z=`"B=:"8:q0!N"C5M=0324!28Z=28B=<@C M=N"0=N"62;7&0=N6< (0, 1) ]7&:µ4:">"2;03<37&28Z=`"B=[\Dt<3B37"2803>UNdZ4!Z3@tT:Q=~F1<33<32;B37dNµT:"030=25Z3@33<=< r CM303<3"Z=<©ª2;B3B3:D B=Nd4!25©qB3:UC57&N` > 1 − α <=62;2879>"<4 α tanh 1 2 log 1 1 + r∗ 1 1 + r∗ 1 √ √ < r < tanh . − x b log + x b α/2 α/2 1 − r∗ 2 1 − r∗ n−3 n−3 B , 1,.23:;,92&%0: 323% ( 2 &/1, % # 23$3: /18 $ 23,.%-$323< %-<3) 18 .1vUC57&` ¯CZ=vw=N"D3B=N&@>"28Z=<3w3<=B=N9<3_kC52~628D=C87&>N90!N"C5M=0324!28Z=28B=<3D%C1MZ=:"7"B3:C87&28D θ) ]3:"M3NQ> Nd@ ;28B32;0=NdZ=`B3v > ξC5:">:"T3vM3B=:C57"`?< x , . . . , x ¯B=28_;N">"<=C8<=6Nd@O>"[\:"03TN?<3_q82;B3280!NdZ=`B3:"D/p(t, C5:>":T3vM3B3:C87&Q@ 62~C57&B3:D{MZ=:"7"B3:C87&`Q> p(u, θ) = Q p(u , θ) ] u = (u , . . . , u ) , .10=24!M3:"Z=:A©ª2;B3<@X:M=N"0=N"62;7&032 "v;4!2~6°B=N&_;[\>N&7"`µC57dN&7&"<37C}@X_~NA4N?@ w=CN J)M303:">2803<=7&`=]w37&:?M303<4N&B=B3[¦|µ_8B=NAw=θ28B3<@| θ >"[\:"03TNC8:" Z!NC5v"2;7"C}@µCS;<3M3:7&28_;:"D<3Z=<B32;7A,^¬FN&TM=0=N&><Z=: C87&03:"<=7"C}@?T=03<37&<3w=2;C5T:2k6B3:A©ª2~C57&>: >qM303:UC57&0!N&B=C87&>"21>C825|_;B=NAw32;B3<3D{M303<3B=<=6N&2~6[¦|>"[\:"03T:D^, <3M3:7&28_~N :7&>"2;035N&2;7"C}@^]=2~CZ="[\"S:03TNqM3:M=NA4N"287q> S <M=03<3B3<=6N"287"C}@R>M303:7&<3>B3:6 C5Z=vw=N&2,.1:~4:""B=:"8: 03:~4)NqC57dN&7&N"287"C}@ @ T303<37"2803<32~6¦, E 7"B3:C8<37&25Z!`"B3:M!N&0=N6287&0=N <=6A2 >c7"C}@µ:C8B3:"Q> @ B=N&@ xzM303:">280U@32;6N&@!k;<3M3:7&28_~N H < w=N"C87&:XT:"SB3T3v03<=03v > ªN&@ x NdZ=`d7&2;03B=N"7&<3>"B!Nd@!S8<3M3:7&28θ_~N H , <=M3:"7&2;_8[ [\>?N >c7SM30=:C57"[6<^] θ = θ ]3<{CZ=:A©qB3[6<^] θ ∈ Θ . ! #! / > \ [ " :"0=TNS<3_FB3:0=6NdZ=`B3:";:ª0!N"C5M=0324!28Z=28B=<@?C1M=N"0=N"62;7&0=N6< (a , σ ) ¯M30=:C57dN&@8<3M=:"7&2;_;N H : > \ [ " :"0=TNS<3_FB3:0=6NdZ=`B3:";:ª0!N"C5M=0324!28Z=28B=<@?C1M=N"0=N"62;7&0=N6< ] ¯CZ!:A©9B!Nd@;<3M3:"7"28_;N=, H : NC8C86:7&03<=6CZ=vw=N"D4>"vU|M303:UC57&[¦|t;<3M3:7&28_,Y.1vC57"` H (a,: θσ =) θa 6=< aH : θ = θ ¯O4!>2?M=03:C87&[\2 ;<3M3:"7"28_8[F,!s TNd©4![6tT303<37&2;03<32~6 C8>&@3_~N&B3[¨:"S<="T3<4!>"vU|03:~4:">=, E ª<3"TNªM3280=>":";:ª0=:~4NS¯q:7&>"2;035N&2;7"C}@;<3M3:"7"28_;N H ]=TU:}4Nª:"B=N>"2;03B=N=, E ª<3"TNª>"7&:03:";:ª0=:~4NS¯qM=03<3B3<=6N"287"C}@{8<=M3:"7&2;_;N ]3T:}4NS>2803B!Nª;<3M3:7&28_~N , O®CZ="280=:A@37&B3:C87&`O:"SH<="T3< M3280=>":";:X03:~4NO< >"2;03:A@37&B3:UC57&`1:S<3T3<ª>7&:"0=:"8:103:~4)N3]&N 1 − β = P (S) ¯S>2803:A@37"B3:C87&`c:7&>"2;038B=vU7"` β = P (S) = 1 − P (S) ;<3M3:"7"28_8v H ]3T:"}4Nª>"2;03B=N8<=M3:"7&2;_;N H , 1 n n k=1 k 1 n 1 1 & 1 1 Hi 1 i i B 1 1 1 1 a KL280=:A@37&B3:C87&` :S<3T3":8:03:~4)N\B=N&_;[\>N"287"C}@Fv03:">B32;6{_;B=NAw3<=6:UC57&<]~Nc>"2;03:A@37&B3:UC57&` ]~0!N&>?@ B=N&@ 1 − β ]U:":"_8B!NAw=N&2;α7"C}@w=28032;_ W (S, θ ) <B=N"_8[\>UN&287C}@q6:SB3:UC57&`N> T303<=7&2803<3@^,!s®0324!<>C528| S @ T3P0=<3(S) 7&280=<328>C v03:">B32;6µ_8B=NAw3":c>C825| S @ T303<37&2;03<32;> C1v03:">B32;6_8B=NAw3UC52;}4NªC8vS2;C87&>"v287A, S∗ U!U1' α Z p(u, θ1 )du . S n1'' r##=' .103254!M3:"Z=:A©q<=6¦]w37&:/M3:/>[\":03T2 B3vU©qB3:X0=N"_5Z=<3w=<37&`O4!>2?M=03:C87&[\2%8<3M=:"7&2;_8[ < ] C5: Z!N"C8B3:TU:7&:"0=[6 >[\":03TNS<=62;287qMZ!:"7&xB3:U,C5.7&.`q. ,0!xN"C5M=0324!28Z=28B=<@ p (u) <Z!"24H2;6 T303<=7&<3w32~C5T:"2u6B=:A©ª2;C87&>": S = {u = (u , . . . , u ) : p (u) > cp (u)} , E 80=N"B3<3w3<=6C}@10=NC8C86:7&032;B3<32;6%C5Z=vw=Nd@^] T:"}4Nk4=Z3@?ZC>c":";: α C5vS2~C57&>v"2;7SM3:UC57&:A@3B3B!Nd@ c 7dN&TNd@^]w37": P (S ) = α , =1'*^ h3±p !$#+ .# !! ' # ! #*# !-, %-, & ) ' # . , + , %&*! ) ! *+ , α # , ## & !+! '&# ! .! # +# #'& !! ' %-# ! H #H'&. S / *! , ,' # ( ' .1vUC57&` S < S ¯T303<37"<3w32;C8T3<32S6B=:A©ª2;C87&>N9C P (S ) = α L4NB=Nd|U:~4!<=6¦]w37&: PH1 (Sc \Sc S) = Z p1 (u)du > c Sc \Sc S Z = cPH0 (S\SSc ) = c Z p0 (u)du = cPH0 (Sc \Sc S) = Sc \Sc S Z p0 (u)du > p1 (u)du = PH1 (S\SSc ) , xzH=, P >"[\M3:"Z=B@32;7"C}@qB3280=N">"2;B=C57&>: p (u) < cp (u) ,&QS:N&><=6RT:"28<=6Ow=NC57~@=6X0=N&>28BL@ ,!K 032;_8v~Z=`d7dN&7&2]M3:&Z!vUw=<=6 S\SSc 7dN&TTN"TqB=N14!:M3:&Z!B328B3<=N{xzH=, P®C5Z!N&5N"2;6:2 P S\SSc Sc H1 (SSc ) 1 PH1 (Sc ) > PH1 (S) . $#rG'j po9n¡#pr#3Upr#n j# $p=()Up#r r#$#!3"o O$#'3#'31U' !#1'j! O#'3"$p jUpr# .1vUC57&` ¯%B328_~N&>"[\":03TNq<3_SB=:"0=6N&Z=`"B3:8:0!N"C5M=0324!28Z=28B=<@RCkM!N&0=N6287&0=N6< .1vC87&` σ <=_8>"2~xC57&B!, .N3.]3.N, x:"7&B=:C5<=7&25Z=`B3:9M=N&0!N"6287"0=N a <=6A2 >c7"C}@4!>"218<=M3:"7&2;_81[ J 8<3M3:7&28_~N H : a = a • 8<3M3:7&28_~N H : a = a > a , • E 7&B3:S2;B3<321M30=N">d4!:M3:~4!:"<@ p (u)/p (u) <=62;2879>"<4 1 n 0& 1 1 1 n p1 (u) n = exp 2 u(a1 − a0 ) − 2 (a21 − a20 ) , u = (u1 , . . . .un ) , p0 (u) σ 2σ (a, σ) , }4!2 u = P u , 11C8B3:=]®w37&: T303<37&<=w32;C8TU:2R6B3:A©2;C87&>": {p (u)/p (u) > c} C5:">M=NA4)N&287/CR6B3:A©ª2~C57&>:6 M303N">"ª<=C5`®7&2;6¦]Aw37&: >"[\":03:"w=B3:"2YC80324!B=282 x 0=N"C8M3032425Z=2;B3:B3:0=6NdZ=`B3: {u > c } CGM=N"0=N"62;7&0=N"6< (a, σ/√n)c ]=B=N&D42;6 :"S<="T3<M32803>:";:"7":"03:8:903:~4NCJ 1 n n i=1 i 1 1 1 1 α = P{x > c1 | H0 } = √ 2π Z∞ c1 −a0 σ e− √ n c1 −a1 σ Z 1 β = P{x < c1 | H1 } = √ 2π u2 2 du = 1 − Φ √ n e− u2 2 du = Φ E :"_8B!NAw3<=6/w32;0328_ u T3>N"B37&<Z!`B=:"0=6N&Z=`"B3:8:0=N"C8M303254!25Z=2;B3<@ −∞ c1 − a 0 √ n σ c1 − a 1 √ n σ xzH=, V xzH=, U γ bFN"M3:6B3"[\7&2;TN&287 c1 − a 0 √ n = uα . σ s¡M3:6:S`Q> x H3, ®B=Nd|U:~4!<=6tT3>N"B37&<Z=` NM3:97dN&"Z=<3321_8B!NAw328B3<=D?1vB3T3=<3< uβ hβ = − Φ(x) c1 − a 1 √ n, σ :M303254!25Z3@32~6:S<3T3v>"7":"03:8:03:~4N β ,QªN&Z=2821C50=N">"B3<=>Nd@ σ σ a0 + uα √ = a 1 − uβ √ = c 1 , n n B=N&|:~4!"[\:"03T=< n ]TU:7&:"0=[\DM303N&2;7k:"ª<3"T=<M3280=>":";:S<>"7&:Q@ 03:8:03:~4N α < β ]=N<=62;B3B3:!] n= σ 2 (uα + uβ )2 . (a1 − a0 )2 1. vUC57&`7&2;M32803`qM!N&0=N6287&0 a <3_;>"2;C87&28B,3N"C8C;6:"7&0=<=6 C5Z=24vI>cS<32G4>"218<3M=:"7&2;_8[1J 8<3M3:7&28_~N H : a = 0, σ = σ • 8<3M3:7&28_~N H : a = 0, σ = σ > σ , • E 7&B=:"S2;B3<321M30=N">d4!:M3:~4!:"<@ 0& 1 1 n 1 1 σn 1 X 2 p1 (u) = 0n exp − x , u = (u1 , . . . , un ) p0 (u) σ1 2 σ02 σ12 i−1 i M30=<3>":~4!<=7ªT{T303<=7&<3w32~C5T:6v?6B3:A©ª2~C57&>v us Z=vw=N"D3B=N&@>25Z=<3w=<3B=N 4!28Z=28B=<@ σ −2 Pn n X x2i > c1 . <=62;287 χ @ 0=N"C8M3032425Z=2;B3<32C n C57&2;M328B3@=6<C5>:"":~4[tC1vB3T33<=28D90=N"C8M3032A@ i=1 2 i=1 xi Kn (x) = 2 Zx 1 2n/2 Γ n n 2 t t 2 −1 e− 2 dt , x > 0 . E S<="T3":";:ª<{>7&:"03:8:03:~4N:"M=0324!28Z3@F> 7C}@<3_F0=N">"2;B=C57&> α=P n 1 X c1 x2 > 2 σ02 i=1 i σ0 β=P n 1 X c1 x2 < 2 σ12 i=1 i σ1 = 1 − Kn c1 , σ02 c1 . σ12 = Kn < β >"[\"<=0=N?>c7"C}@>q_;N">"<=C8<=6:C87&218<3M3:7&28_;[1J KL25Z=<=w3<3B3[ α 1 • H0 : p0 (k) = Cnk pk0 (1 − p0 )n−k & , E M37&"2;03T3 H1 C57"03:"<37C}@?<=C}|U:~4=@<3_FB=280=N">"28B!C57&>UN p1 (k) h p1 (1 − p0 ) ik (1 − p1 )n = > c, p0 (k) p0 (1 − p1 ) (1 − p0 )n T:"7&:03:"2S0=N&>B3:C8<Z=`"B=:B3280!N&>"2;B=C57">"v k > c M303<B=28T:"7&:03:6 c ,!QkZ3@µ>"[\w=<=CZ=2;B3<@µ:S<3:"T%M32;03>":8:"7&:Q@ 03:8:p0=:~4N>:C8M3:&Z=`_8v2;6C}@/N"C5 C5Z=vw=N&D=B3:"D>25Z=<=w3<3B3[ µ CM!N&0=N6287&0=N6< , F6282~6 (np, np(1 − p)) 1 1 n n µ − np o c1 − np0 n α = P{µn > c1 |H0 } = P p H0 , >p np0 (1 − p0 ) np0 (1 − p0 ) n µ − np o c1 − np1 n 1 β = P{µn < c1 |H1 } = P p

S<=C8`S:"ª<3"T:DM32;03>":8:03:~4N ]=B=N&D42;6<=_FM32803>:";:ª0=N">"2;B=C57&>UNST=>N&B=7&<Z=` B=:"0=6N&Z=`"B3:8:0=N"C8M3032A@ 4!28Z=28B3<3@tCM=N"0=N"62;7&0=N"6< (0, 1) 7&:"03:8:%0=N">"2;B=C57&>UN{B=Nd|U:~4!<=6 u": ª<3"T=vR>7&:"03:8:%03:~4)N β QkZ3@?>"280=:A@37&B3:C87&28D?:"ª<3":T α < β 80!N&B3<3!N c M30324)C57dN&>"Z3@3287"C}@?>9>"<4!2 α 1 1 c1 ≈ np0 + uα p np0 (1 − p0 ) ≈ np1 − uβ p np1 (1 − p1 ) . E 7"CA>L4NB=Nd|U:~4!<=6B=28:"A|U:~4!<=6[\DM=03<3&Z!<3_8<37"25Z=`B3[\D{:"QP2;6>[\":03T3< n≈ uα p p 2 p0 (1 − p0 ) + uβ p1 (1 − p1 ) , (p0 − p1 )2 5N&0!N&B37&<=03v > S<=D:S<3T3vM32;03>":8:03:~4N α <{:"ª<3"T3vq>7&:"0=:"8:03:~4N β , + A! 2&: 3$&%-,98 % &% 32 1$ 7/!418 .1vUC57&`C8:C87&:A@3B3<32C8<=C57"2;6[E:"M=<=C5[\>UN&287C}@ 7&:w3T:"D FN"_8:">:";:OM=03:C87&0=N&B!C57&>UN ,QªN?@ Z=2;2C5:UC57&:A@3B=<32C5"03=2~6#{e@ t , 4. .<=.C5,T3e0=28}7&B3:R< M30=<3B3<=6N"287_;B=NAw32;B3<@ 0, 1, . . . , T ] T 6 +∞ ,£®>:&ZC>c3<@<3_;vw=N&2~6:"DtC8<=C57&2~6[ :"M=<=C5[\>UN&287C}@X7"0=N&2;T37&:"0=<328D ]¦}4!2 ω = i ]L2;C5Z=< > 6:628B=7>"032~628B3< t C5<=C87&2;6NtB=Nd|U:~4!<Z!NC5`/>tC8:C87&:A@3B3<3 ; 2 3 A : 3 @ & 7 3 B :UC57&B3:29M303:C87&0=N"B=C57&>:=]^>%T:7&:"03:U6 Ω = {ω} ¯M=03:C87&0=N&B!C57&>: =5Z=2~628B37dN&0=B3[¦|OC8:""[\7"<3D^] (Ω, U, P O ¯ & N = Z 8 ; 2 " = % N U > 5 C 8 2 R | 3 M :~46B=:A©ª2;C87&>=]p>"2;03:A@37&B3:UC57&` _;Nd4N&2;7"C}@ =5Z=2~628B37dN"03B3[6<>2803:A@37&B=:C57~@=6< ,KL>"2R@ 4!U2~6¡M3:UCZ=254!:">UN&7&28Z=`"B3:UC57&`CZ=vw=N"D3B3[¦|µ>"28Z=<3w3<=B {ξP } ]!M=:&Z!N"5Nd@ ξ (ω) = ω M3:&Z!:A©ª28B=F< >yC8<=C57"2;6[P {ω} >{6:62;B37 >032;62;B3< ,^KL280=:A@37&B3:C87&< 6:"8v7?_;N">"<=C8287&`:"77":"8:!])>TN&T=<|C5:UC57&:A@3B3<3@|µ"[¦Z!NC5<=C87&2;6N?>?M303R2 @ 4![L4vUª<| t (n − 1)@ 6¨"2;03:A@37&B3:UC57&N"B3B3:U6 <=C}|U:~4!2 ω B32_~N&>"N"7&25Z!`"B3:UC57&`C5Z=vw=N&D=B3[¦|%>"25Z!<3w3<3B {ξ } B=N"_8[\>N"287"C}@ <=C}|U:~4!:">q:UC57dNdZ!`"B3[¦| 32;M3Q` > µN"03T:">UN (t − 2) 1 T r t t t t t t t P {ωt = i|ω0 = i0 , ω1 = i1 , . . . , ωt−2 = it−2 , ωt−1 = j} = P {ωt = i|ωt−1 = j} = ptij . SLv;4!2;6 M30=24!M3:"Z!N&5N"7&`=]Pw=7&:µ>µB=NAw!NdZ=`B3[\DO6:U628B37>"032~628B3 S<=2F325Z=:w3<=C5Z=28B3B=[\2ª6:62;B37&[°>"0=2;628B=<C5<=C87&2;6N628B@32;7 ω : P {ω = j} = p , C8>":"2qC8:C57":A@3B3<32"]M=03< 8= 7&:U6 >p2803:A=@37"1B3:C87&`{>µ6:628B=7%>"032~628B3< t 4=Z3@C5<=C87&2;6[yM3:M=N"C87&`%>%C5:C87&:A@3B3<32 ω = i _~N&>"9TN&T:6 C5:C87&:A@3B3<39M30=24![L4!vª<3D%6:628B=7>032;62;B3<^, ! # ! x ®S Z=vU©ª4)N&B3<32FCGM3: Z=:"ª28B3<32~6c6:U628B37 t w=N"C87&<3=NcB=N&|:~4!<37C}@1>cM3:&Z!:A©ª28B=<3< k ] 0 < k < a ];7":c>cM3:C5Z=24!v c> S<3D 6:U628B37%>032;62;B3"280=:A@37&B3:C87&`Q> M3280328|:~4!<=7?>%M=:&Z=:A©28B3<32 <Z=< , ®O CZ=< <Z!< ]7": w=NC57&<==N{:C87AN"287"C}@X>%7&:w3T2 k Cq>"2;031/2 :A@37&B3:UC57&`N> 1 , M N"TU:D/M303:32;C;C9k"−v;4!12;6 B=N"_8k[\>U+N&7&1`{&Z!v©4kN"=B3<32~06 C9M=:"a Z=:Q@ ª28B3<32~6¦,.1vUC57&` ξ ¯?M=:&Z=:A©28B3<32ªw=N"C87&<33[ >?6:62;B37>032;62;B3< n ,.1:C5Z=24!:>N"7&25Z=`B3:C87&` {ξ } :""0!N&_8v287 32;M3`qµN&0=TU:>N=, M N&T3<=6 "Z=vU©ª4)N&B3<32~6t6:A©qB3::"M3N"7&` "2;_8:"<4!B=vI> \<3;03v",!.1:"M=Nd4N&B3<=2c> 0 <Z=< a "v;4!2;7 :_8B=NAw=N"7&`0=N&_;:"032;B3<32G:~4!B3:";:ª<=_F<3803:T:">=, ®O CZ=<µw!N"C57"<3=N<=_S7&:w3T3< CS>"2;03:A@37&B3:UC57&`N> M3280=25|:~4<37q> ]!Nq<3_ªM3:&Z!:A©ª28B=<@ a CF>2803:A@37"B3:C87&`Q> 1 >97&:w3T3v a − 1 ]7&:q&Z=vU©4N"B3<321B=0N&_;[\>N"287"C}@"Z=vU©4N&1B3<=2;6tC1:7&0=N&©ª281B=<32;6¦, ®O CZ=< B32c_~N&><=C}@37F:"7 ]7&:ª328M=`ªµN"03T:">NkB=N&_;[\>N&2;7"C}@q:~4!B3:03:~4!B3:D^, M :"Z=`"T:ª7AN"T3<32c328M3N v842;6>ª4N&Z=p`"B=28D3ª2;6 0=N"C;C86N&7"t03<3>N"7&`=, >"2;03:A@37&B3:UC57&< :"0=N&_;v > 7 6N"7&03<3=v C8: QkZ=@ T:"B32;w3B3:"D¨328M=<¨µN"03T:">N¨x C8>":"D!C57&>UN"6 0 UTC528<| i∞) ,^N&7&03<=3[CvTN&_;pN"B3B3[6":D=C57">N"6NQ=k > 7"C}p@C87&k:A|UN?@ C87&<3w32~C5T3<=6<, KL>24!2~6 :":"_;B=NAw32;B3<@ J p (k) = P {ω = j|ω = i} < [P (k)] =k p (k) k , 1_µ1:0=6v~Z=[ M=:&Z=B3:D j r j=1 j t ∞ n n=0 n t ij ij ij >2803:A@37&B=:C57"L4)NC5Z=24v"287A]3w=7&: s=1 [P (k)] = [P (k − 1)] [P ] s=1 k P {ωk+n = j|ωn = i} = P {ωk = j|ω0 = i} . =+ < pij (t + s) = M N"T3<=6 :"0=N&_;:6¦]32;M3`µN"03T:">N r X pik (t)pkj (s) . |N&0=N"T37&2;03<3_8v287"C}@?>"2;T37&:"0=:6 B=NAw=NdZ!`"B3[¦|>2803:A@37&B=:C57"28D (p , . . . , p ) < 6N&7"03<332;D [P ] M32;0325|U:~4!B3[¦|>2803:A@3{ξ7&B=}:C57"28D{C =5Z=2~628B37dN"62803:A@37"B3:C87;@|=5, .! # ! '& ! t .# # #' ,' ! ' # ( $# '& )& #' ) #' .! # +# lim p (t) = p # j = 1, . . . , r / !$#+ # ( # # ! ' [P (t )] k=1 1 t ij ' pj *! , ,' # ( ' KL>24!2~6:":"_;B=NAw328B=<@ / i r X J xj = 1 . mj (t) = min{pij } . i mj (t) 6 pij 6 Mj (t) . .1:q1:"0=6v~Z=21M3:"Z=B3:D>2803:A@37&B=:C57"< pij (t + 1) = X pik pkj (t) . ]3B3287"03v;4!B3:q>"<4!287&`!]w37&: k [P ] mj (t) 6 pij (t + 1) 6 Mij (t) . E 7"CA>L4NB=Nd|U:~4!<=6 .1:Q=;7&:6vFC8vUª2;C87&>"v > 7 M3032425Z=[ < 7dN"TU:>"[F]w37&: i j mj (t) 6 mj (t + 1) 6 Mj (t + 1) 6 Mj (t) . limt→∞ mj (t) pik = Mk (t + t0 ) KL:C8M3:&Z=`_8v2;6C}@M303254C57dN">&Z=2;B3<@=6< < j j=1 i W w=<37&[\>N&@{C87&:A|UNC57&<=w3B3:C87&`96N&7&03<=3[ ij j = 1, . . . , r , Mj (t) = max{pij }, 11C5B=:=]w37&: ' #+' .# !$#% # # ' # t→∞ xk pkj = xj , k=1 t # ,. ' ' ,"%-, ( ") '& r X t+1 r < limt→∞ Mj (t) < ,;QS:"TN&©ª2;6¦];w37&:GM303254!25Z=[C8:">M=NA4NQ> 7A,A.1vC57"` pjk (t + t0 ) = mk (t + t0 ) . Mk (t + t0 ) = pik (t + t0 ) = r X pi` (t0 )p`k (t) `=1 mk (t + t0 ) = pjk (t + t0 ) = KL[\w37&2~6t<3_FM=2803>:"8:0=N">"28B!C57&>UNS>7&:"03:2 Mk (t + t0 ) − mk (t + t0 ) = r X pj` (t0 )p`k (t) . `=1 r X `=1 pi` (t0 ) − pj` (t0 ) p`k (t) . U I N&_;:"`"2;6?C5v66v P B=NC5v66v Mk (t + t0 ) − mk (t + t0 ) 6 Mk (t) M N&T{TN&T 0= Pr `=1 :7&03<3=N"7&25Z!`"B3[¦|FCZ!N"5N&2~6[¦|), M :}4N X x P!, +A (t ) + m (t) p (t ) − p (t ) . B328:7&03<3=N"7&25Z!`"B3[¦|kCZN&5N&2~6[¦|c< P+ X+ ` pi` (t0 ) − pj` P+ P− pi` (t0 ) − pj` (t0 ) = + dij = 1_1vC5Z=:"><3D7"28:"0=2;6[°CZ=254!v"2;7~]=w37&: X+ i` , E ":"_;B=NAw3<=6w328032;_ j` ` P− =− P+ d = max dij < 1 dij C5v66v , 1_qx P=,-+~L7"28M32;03`9M3:&Z=vw=N"2;6 Mk (t + t0 ) − mk (t + t0 ) 6 Mk (t) − mk (t), M :"}4N Mk (t) − mk (t) 6 d 3M 0=< t ∈ ∞ , E 7"CA>L4NC5Z=24v"287?C5vS2~C57&>:">UN&B3<32SM3032425Z!N QªN&Z=282]U>9v0=N">"B32;B3<3< pij (t + 1) = M32;0328D4!2;6 T{M3032425Z=v<{M3:"Z=vw3<=6 t t0 →0 lim pik (t) = pk r X r X < pk pjk , j = 1, . . . , r, r X xk pkj (t + 1) = .12;0325|U:~4=@T?M3032425Z=vM303< k=1 M 28:"0=2;6Nª4!:TN"_;N"B=N3, , j=1 xj = r X xk pkj (t) k=1 r X xk r X pk` p`j (t) = ]3M3:"Z=vw=N&2~6 t→∞ k=1 `=1 r r X X `=1 xj = r X r X xk pk` p`j (t) = x` p`j (t) = xj . k=1 `=1 xk p j = p j . k=1 #%-, # / K¡w=NC57&B3:UC57&<]4!:"TN"_;N&B=:=]w37&: 1_10=N">"28B!C57&>UN x P!, I pj = 1 . v;4!:">"Z=287">":"0U@F>c7C8<=C57&2~62Fv0=N&>B328B3<=DOxzP=,JI", 1B4!vT33<32;D4:"TN&_;[\>N&2;7"C}@^]w37&: M30=<ZK> ":U6 t ,=K C8N"6:U6X4!28Z=2"] r X mk (t) 6 pik (t) 6 Mk (t) pik (t)pkj k=1 x1 , . . . , x r ]!7dN"TTN&T t→∞ k=1 pj = .1vUC57&` − k pi` (t0 ) − pj` (t0 ) . ]M3:Q=;7&:6v <1 ` dij ]37&: P− |pij (t) − pj | 6 d pi (t) = P {ξt = i} = t t0 r X k=1 . p0k pki (t) , H vw3<37&[\>UNd@^]pw37&: M303< ]M=:&Z=vw=N"2;6 p (t) → p <M30=24!28ZB32_~N&>""2;03:A@37&B3:UC57&2;D^, p (t) → p t → ∞ s®:UC57&:A@3B3<=2 i ∈ E B!N&_8[\>UN&2;7"C}@B32;C8vS2;C87&>"2;B3B3[6¦]2~CZ=< C5vS2~C57&>v"287R7dN&T:"2 j ∈ E < 325Z=:2 t ]w37": ] 4=Z3@ZK> ":8:9328Z=:";: , p (t ) > 0 p (t) = 0 .1vUC57&`9C5<=6>:&Z S :":"_8B!NAw=N&2;7kTZ!NC8CGtB32;C8vS2;C87&>"2;B3B3[¦|C8:C87&:A@3B3<3D^,QS:M3:&Z=B=<37&25Z!`"B3:2G6B3:A©2;C87&>":9C5Q: @ C57":A@3B3<3D B=N"_8[\>N"287"C}@TZ!N"C;C5:U6C5vS2~C57&>28B3B=[¦|C5:C87&:A@3B3<3D,3s®vª2;C57">"28B=B3[\2GC8:C87&:A@3B3<@ < B=N"_8[\>?N @ > 7C}@{C8:":"E\S ªNQ> S"v > 797dN&T=<321325Z=[\2Fw=<=CZ!N t > 0 < s > 0 ]w37&: p (t) > 0i 0 , E 7&B=:"S2;B3<32C5:":"N?> ª<32;C}@?@3>&Z=@3287"C}@RC8<=66287"03<3w3B3[6¦]0=2;GZ=2;TC8<3>"B=[6<7&0=N"B3_8<=7&<3>"B=[6¦,)µB3:A©ªA2 @ C57">":XC5vS2~C57&>28B3B3[¦|tC8:C87&:A@3B3<3D 0=N&_;"<3>UN&287C}@B=NT:B328w3B=:"2<3Z=< C8w3287"B3:"2{w3<=C5Z=:OB328M32;032;C828T?N >cS<|C}@t6B3?: @ ©ª2~C57&> E , E , . . . <3_C8:":"ªNQ> ª<|UC}@XC5:C87&:A@3B3<3D,p%B3:A©2;C87&>N E , E , . . . "v;4!2;6 B=N"_8[\>UN&7&`%TZN"C8C;N"628B3B=[¦|C8:":"ªNQ> S<3|UC}@ C8:C87&:A@3B3<3D^,L128M3`N&03T:">UN3]uC8:C87&:A@3ªN&@ <3_:~4!B3:";:/TZ!NC8C8NtC5vS2~C57&>28B3B3[¦| C5::"ªN?>cS<|C}@C5:UC57&:A@3B3<=D^]B=N&_;[\>N"287"C}@B=280=N"_5Z=:A©q<=6:D^, KL>"254!2;6t:":"_8B!NAw328B3<3@ J ki i i i ij ji ij 1 2 1 ji 2 fj (n) = P {ξn = j, ξn−1 6= j, . . . , ξ1 6= j|ξ0 = j} Fj = ∞ X fj (n) . !4!2~C5` ¯%>2803:A@37"B3:C87&`=]w37":C5<=C87&2;6N=]^>"[\D4=@O<=_ @ 8:%C8:C57":A@3B3<@^]^>M32803>[\29>"280=B3287"C}@X>{B328;:%B=N @ :6 ª N"82] Ff (n) ¯>"280=:A@37&B3:C87&`=]w37":C5<=C87&2;6N=]3>"[\D4!@<3_ j @ 8:qjC5:UC57&:A@3B3<3@^]>"B3:>"`ST:}4N @ B3<3v84`ª>2803B32;7"C}@>9B32;n8:!, M 28:"0=2;6Nq:qM30=24!28Z=`"B3[¦|?>"2;03:A@37&B3:UC57~@|6:A©ª2;79"[\7&`qvUC5<Z=2;B=N3,)s®:C57":A@3B3<32 B=N"_8[\>UN&287C}@M32;03<3:?@ 4!<3w=2;C5T=<=6 CSM32803<=:~4!:6 d ])2;C5Z=<>:"_8>0=N&ª28B3<=2kCSM3:"Z=:A©q<37&28Z=`"B3:Dµ>"2;03:A@37&B3:UC57&`N> j>?∈C5:EC87&:A@3B3<32 j >":_;6:A©qB3: 7&:"Z=`"T:_~Nµw3!]T30!N&7&B3:2 d < d @3>"Z3@3287"C}@ :""ª<=6°B=N"<3":"Z=`"ª<=6 4!25Z=<=7&25Z=2~6 6B3:A©2;C87&>Nw3<=C528Z , {n : f (n) > 0} OuC5Z=<C8:C87&:A@3B3<@B32;0=N&_8Z=:A©q<=6:"D328M3<µN&03T:>NG¯9M32803<=:~4!<3w32~C5T3<32GCM32;03<3:~4!:U6 d > 1 ]7":ª32;M3`SB!N&_8-[ @ >N"287"C}@RM32803<=:~4!<3w32~C5T:"D^H, OuC5Z=UNw3 0} 0=N&>28B+ 4=Z3@{B32;TU:7&:"0=:"8: j ]332;M3`9B=N&_;[\>N&2;7"C}@B32;M3280=<3:~4!<3w32~C5T:"D, =1'±pz ) ' ( % '& # , ! ! ., ! #% '&*) # % ' .! & j ) #' ., !$#+ # lim p (t)N= p # j = 1, . . . , r # # .,.. & ' / i ## "+ +' ,' *% # % '& #( ,*# # ! "+ %-#! !$#+ ' ,. , "+ ! ,. )& #' .# '& ! / n=1 j j j j j j j t→∞ '& ! # + # ( # .# ! , '&' ij j pj r X '& # +'&.# !$#% ## ' # xk pkj = xj , k=1 ! #'&") # ) #'&.) #' '&, ! # j = 1, . . . , r , r X xj = 1 . j=1 # % '& G1<=C5Z!N p @3>&Z3@F>c7"C}@R>2803:A@37"B3:C87;@=6 j @ 2qC5:UC57&:A@3B=<329w328032;_q":"Z=`"ª:"DµM30=:62R@ ©qv7&:"T¡>032;62;B3<^,¦.103< 8= 7&:U6 :"TN&_;[\>N"287"C}@^]w37": 8= 7&< >"280=:A@37&B3:C87&<¡B32µ_~N&><=C}@37/:"7/B=NAw!NdZ=`B3:";:XC8:C87&:A@3B3<@ h<1 |pij (t) − pj | < ht j C5N&2;7"C}@>:"_8>0=N&7"B3[6¦]2;C5Z=< F = 1 <{B328>:"_;>"0=N"7&B3[6¦]2;C5Z=< =1'±p h^ '&, # j ∈ E . !-,'& ' "),+ , ' (! '&"),+ ,*# ! "),+ , j Pj = #. !-,'&") &'&, *! , ,' # ( ' .1vUC57&` / ∞ X n=1 Fj < 1 , pjj (n) = ∞ . j∈E Fj = fj (0) = 1 Pj . 1 + Pj ,3.1:91:0=6v~Z=21M3:&Z!B3:"D{>"2;03:A@37&B3:UC57&< pjj (n) = fj (1)pjj (n − 1) + fj (2)pjj (n − 2) + · · · + fj (n − 1)pjj (1) + fj (n) . x P=, H "P LK >"254!2;6tM303:<3_8>:~4=@3S<3211vB3T33<=<M3:UCZ=254!:">UN&7&28Z=`"B=:C57"28D Pj (z) = ∞ X pjj (n)z n , pjj (n) Fj (z) = n=1 WG6B3:A©N&@Ox P!, HU®B=N zn <{C8TZ!Nd4![\>N&@M303< ]3M=:&Z=vw=N"2;6 n = 1, 2, . . . < fj (n) ∞ X fj (n)z n . n=1 Pj (z) = zfj (1) 1 + Pj (z) + z 2 fj (2) 1 + Pj (z) + · · · = Fj (z) 1 + Pj (z) . x .103< >C82CZ!N"5N&2~6[\2ª>xzP=, HGB32;:"7&03<==N&7&28Z=`"B=[F]^M3:Q=;7&:6v803vM3M3<=03:">UN&7&`{<|X6:A©ª2;6 ZC>c"[6¨:"?@ 0=N"_8:U6¦x,p.1∈:µ(0,7&2;:"1)032~62¥ "25Z=@X0!N&>"2;B=C57">":µC5:A|0=N"B@3287C}@Xµ24!<=B3<3w3B3:U6°T303v82B=N%T:6M3Z=28TC5B=:"D/MZ=:UC5T:C87&<^,J M N&T3<=6 :"0=N&_;:6¦] P (z) = F (z) 1 + P (z) , E 7"CA>L4N9B=N&|:~4!<=6 j j j Pj (z) = Fj (z) , 1 + Fj (z) Fj (z) = Pj (z) . 1 + Pj (z) 1. vUC57&` P = ∞ , M :}4N P (z) → ∞ M30=< z → 1 − 0 ,.1:N=87&:U6v F (z) → 1 , M N&TTN&T F (x) < F 6 ]M3:&Z!vUw!N&2;6 , 1, x ∈ (0, 1) .1vUC57&`S7&28M32;03` F =F1 , =M :"1}4N F (z) → 1 M303< z → 1 − 0 , E 7"CR>L4NªCZ=254!v"2;7A]Uw=7&: P (z) → ∞ ]NS_;B=NAw3<37A] ,3bFN"T:"B32;^]2;C5Z=":_8>"0!N&S2;B3<3Dª>1C8:C87&:A@3B3<32 j 0=N">"B3: P j I , .1N: =87&:U6v j j j j j j j j j ∞ n=1 j ∞ n=1 j j jj j Pj 1+Pj jj M ∞ n=1 {ξn =j} ∞ hX ∞ ∞ i X X I{ξn =j} = M I{ξn =j} = pjj (n) = Pj . !#1'%± h x C5:&Z!<4N&0=B3:C87&Nq0=N"_5Z=<=w3B3[¦|%C5:C87&:A@3B3<@^,s®vS2;C87&>"v >c7qw3<=CZN N < M 7dN&T3<32]!w=7&: n=1 ] / pkj (N ) > 0 pjk (M ) > 0 n=1 n=1 ,=.1:1:0=6v~Z=2FM3:"Z=B3:"D?>"2;03:A@37&B3:UC57&< pkk (N + M + n) = E 7"CA>L4N9CZ=254!v"2;7A]3w37&: X pk` (N )p`s (n)psk (M ) . `,s pkk (N + M + n) > pkj (N )pjj (n)pjk (M ) = αβpjj (n), }4!2 n ¯kZC>c":"2] α = p (N ) > 0 < , E 7"RC >L4)NªM=:&Z=vw=N"2;6 αβp (n) kj β = pjk (M ) > 0 , M :"w=B3:F7dN&T9©ª26:A©qB3:1M3:TN&_;N"7&`=]dw=7&: pjj (N + M + n) > kk 1 pkk (n + M + N ) > pjj (n) > αβpkk (n − N − M ) αβ 4=Z=@ n > N + M ,.1:Q=;7&:6vqN"C8<=6M37&:7&<3w32~C5T3<=21C8>":D=C57&>UNkM3:UCZ=254!:">UN&7&28Z=`"B=:C57"28D p (n) [F, KXw=NC57&B=:C57"<^]&2;C5Z= αβ P p (n−M −N ) = ∞ , M N&T3<=6 :"0=N"_8:6¦]2~CZ=<{C5:UC57&:A@3B3<=2 k ∈ E >:"_;>"0=N"7&B3:=]7&:q<{C5:UC57&:A@3B=<32 j ∈ E >:"_8>0=N&7"B3:=, &j#lm'n#po 3jl '3pr# $#O/U^ k ∞ ∞ kk n=1 n=M +N +1 jj ∞ n+M +N +1 kk kk jj U V .103254C57dN&><=6¡w=NC57&<=3v"]^T:7&:"0=N&@%4!><©ª2;7"C}@M3:{328Z=[6 7&:w3TN"6 B=N?M30U@=6:"D^,^.12;032;62;S28B=<32ª<=_q:~4!B3:"DR7&:"w=T3< >?4!03v;vI> M303:""B=[\2qM303:628©qvU7"T3"032~628B3<,#N{:~4!<=B/ªN&qw!N"C57"<3=N{<3_7":"w3T3< k CqM3:"Z=?: @ ©q<37&28Z=`"B3:D>2803:A@37&B=:C57"`Q> p M32;032;62;ªN&2;7"C}@>7&:"w3T=v k + 1 "2;03:A@37&B3:UC57&N` > q = 1 − p M32;032;62;ªN&2;7"C}@>97&:w3T3v k − 1 ,S<3_;<3w32~C5T:"D{C5<=C87&2;62kC8:":7&>"2;7"C57&>v"2;7ª32;M3`qµN&0=TU:>N ( CG>"2;03:A@37&B3:UC57&N` > p 1 ξ =ξ +η =ξ +S , η = CG>"280=:A@37&B3:C87&Q` > 1 − p . −1 N6287&<=6¦]=w37&:q>"2;03B3v7&`UC}@>9TN&T3v > @ Z=<3:ª7":"w3T3v6:A©9B=:7":&Z=`TU:9_~Nªw=287&B3:2Fw3<=C5Z=:9ªN&;:">!, =1'±p-* )%-, # ). +,.# , .! !-,.) #' !-,' ) #( , ! ! ,*'&*),+ , n n−1 n n n '& ( ! ' "),+ ,*#! "),+, p = q = 1/2 / "Z=vU©4N&B3<=2@3>&Z=@3287"C}@B32;0=N&_8Z=:A©q<=6:"D/=28M3`N> µN"03T:">UN3,.1:R7&28:032;62 *! , ,' # ( ' .10=< C5:"Z=<4N"03B3:UC57&<4!:C8/ 7dN&7&:w3B30:9<<3_8pvw3<<=7&`91 7&<3M%:~4!B3:D7":"w3T3<, s =87&:D9325Z=`N><=C8C5Z=24!v2;6OC}|:~4!cS2;DOw!N"C57"<33[yM3:C5Z=2 N&8:>=]7&: S CZ!vUw!N&D3B3[¦|>"25Z!<3w3<3B{0=N">"B3[ 1 ]3N:C87AN">"ª<32;C}@ k 0=N&>B3[n −1 ,3.1:Q=87":6pv ∞ n=1 00 n k k P {S2k = 0} = C2k p (1 − p)k ∼ √ .1:Q=;7&:6vM303< <{0U@4 p 6= 1/2 0U@4 P∞ k=1 (k − 2kp)2 1 exp − 4kpq 4πkpq p00 (2k) C}|:~4!<37C}@^]NM30=< p = 1/2 00 00 (2k) = P {S2k = 0} ,^KC5v662 S2k k (1 − 2p)2 1 1 = √ √ exp − k 2 pq πk 4pq <=6282~6 1 p00 (2k) ∼ √ πk =0 N"C}|U:~4!<37"C}@^, , $, # %-,.# .103< N&5C5N\0=2w!4!N"B3C52;7"2k<3=w3N\<=C5B3Z=<=:TU:>}:"4_8N\>0=N&:&ªZ=`28SB3<=2D?B32®>>"B328v~0=Z=B3`287"TC}:"@FB3>c2;w3B3B3v~:!Z=]=`=:", B=GF:qN"C50=7"N&<3>3B3vF: C5PB32~C52;7p>"M30!(2k) £®7&:c_8B=NAw3<=7~]~w37&:GC/ B32;TU:7&:"p0=:"6=8:\1/2 N&>":!] 2;C5Z= 1/2 <Z!"Z=28>:=]2~CZ=":_8>0=N&ª28B3<3D9>kB=v;Z!`S"2~C5T:"B=28w3B3:!]&:"B=:k0=N"C87&287A]UB3:kB32cM303:M3:"03=<3:"B=N&Z=`"B=: .103< w3<=C5Z=vN&;:"p> = 1/2 1 < X p (2k) ∼ X √1 ∼ 2√n . p (2k) ∼ √ P∞ k=1 p00 (2k) ∞ k=1 n 00 πk 00 n 00 πk π M N"T3<=6{:"0=N"_8:6¦] >\28_8:"<4B3:"DG<380=2#4!:&Z3@FB3<=w3`"<|G0328_;v~Z=`d7dN&7&:>¦Cv>25Z=<3w=28B3<32~6w=<=CZ!NªN&;:">L"[C87&03:\M=Nd4N&2;7~] N:"7&T3Z=:"B32;B3<32FM3:9>25Z=<3w=<3B321<4=Z=<37&28Z=`"B=:C57":"_;0=N"C87ANQ> 7A, k=1 k=1 K 4!>v62;03B3:U6 CZ=vw=N"D3B3:U6 &Z=vU©4N&B=<3< w=NC57&<3!NM32;032;62;ªN&2;7"C}@ 24<3B3<3w3B=[6< ªN"5N"6< > w3287"[\0325| =B N"M30=N&>"Z=28B=<@|M=N"0=NdZ3Z!25Z=`B3:ª:UC}@=6 x < y ,UQkZ3@{w=N"C87&<33[F]3>[¦|:~4=@3ª28D<3_FB=NAw=N&Z!NST::"04!<3B=N"7A]>":"_~6:A©qB3[6< M3:"Z=:A©ª2;B3<@=6"0!N&S2;B3<32%>B!NAw=NdZ=`B3:"2%M=:&Z=:A©28B3<32%>:"_~6:A©9B=:/7&:&Z=`T:/7&:"}4N=]uT:"}4)NXTU:"Z=<3w32~C57&>: / ªN"8:>>OM3:&Z=:A©q<37"25Z=`B3[¦|B=N&M=0=N&>"Z=28B3<3@|X:UC52;D x < y 0!N&>"B3[ C5:":7&>"2;7"C57">"28B=B3:T:&Z!<3w32;C87&>N6 ªN"8:>>O:"7 @ 03<3!N&7&28Z=`"B3[¦|OB=N&M=0=N&>"Z=28B3<3@| =87&<3|R:C828D^,p.1Q: =87":6v p (n) = 0 ]2;CZ!< n B328w=287&B3:!]^<¡x >:C5M=:&Z=`_8v"2~6C}@_42;C5` M3:"Z=<3B3:U6<=NdZ=`B3[60!N"C5M=0324!28Z=28B=<32;6Y 00 p00 (2n) = n n 1 n X k 2 1 X (2n)! Cn . = C 2n 42n k!k!(n − k)!(n − k)! 42n k=0 k=0 GG Z=2;B3[ ;<3M32;0382;:6287"03<3w32~C5T:";:0!N"C5M=0324!28Z=28B=<@>qC5v662G4!:"Z3©qB3[ 4N&>UN&7&`24!<=B3<33v,=.1:Q=;7&:6v M N&T{TN&T x P!, P n Ca0 Cbn + Ca1 Cbn−1 + · · · + Can Cb0 = Ca+b . Cnk = Cnn−k ]3<3_qx P!, P¦CZ=254!v"2;7A]w37&: Cn0 K¡0=28_8v~Z=`d7dN"7&2"]M3:"Z=vw=N&2~6 2 + Cn1 2 + · · · + Cnn n p (2n) = 4−2n C2n s M3:U6:"ª`Q>¨1:0=6v~Z=[¨s®7&<303Z=<3B35NSB=Nd|U:~4!<=6¦]Uw37&: _;:60U@4 P p (2n) 0=NC}|:~4!<=7"C}@^, K 7&0325|362;03B3:6C5Z=vw=N&2FN&B!NdZ=:8<3w3B=:ª<=62;2;6 00 n k=1 p00 (2n) 2 2 n = C2n . . @3>&Z=@3287"C}@>25Z=<3w=<3B3:"DM3:"0U@4!TN 1/n ]7dN"T3<=6X:"0=N @ 00 p00 (2n) = 6−2n X j,k>0 j+k6n o2 n! 1 n X n1 (2n)! = 2n C2n . j!j!k!k!(n − j − k)!(n − j − k)! 2 3n j!k!(n − j − k)! j,k>0 j+k6n KLB3v7&03"B=N%+,.1:N=87&:U6v%C5v66N<| T3>UNA4!0!N&7&:>ªB321M30=28>":UC}|:~4!<=7ª6N&TC8<=6NdZ=`B3:";:ª<=_FwZ=28B=:">>C5T:"TN&|,UK¨C8N"6:U6X4!28Z=2"]=2;CZ!"2;7"C57">"v > 79_8B!NAw328B3<3@ < ]=0=N&>B3[\21M303<=62;03B3: ,!s¡M3:U6:"ª`Q>1:"0!6v;Z![ s®7&<=0Z=<3B35N 3B 2~CZ=:A©qB3:Xv"254!<37&`UC}@^]w37&: =;7&:"76N"TC5<=6jv6ykC50=N">"B3<=6yC 1/n ,Y.1:Q=87"n/3 :6vt>2803:A@37"B3:C87&` p (2n) C50=N">"B325Z=<3w=<3B3:"D 1/√n ]37dN&T?w37&:q0U@4 P p (2n) C}|:~4<37"C}@^, ikj#lm'n## =j#l Y'#r#%$X²"joq 3=m#^'3 ! U' [0, n] E UC5v;4!<=6 "Z=vU©4N&B3<=2µB=NM=03<=6280=2_;Nd4NAw3<¡: 0=N&_;:"032;B3<3<^,L.1vC57"`B=NAw=N&Z=`"B3[\D¡TN"M3<37dNdZ M32;03>":8: <3;03:"TNC8:C87AN">&Z3@32;7 k 0=vU"Z=28D]NTN&M3<37dN&Z>"7&:03:";:<3;03:"TN9¯ (n − k) 03v&Z=2;D^,).12803>[\Dµ<380=:"T>[\ξ <380=[\>N&2;7 <Z!<µM303:<3803[\>UN&2;7q03v&Z=`CS>"2;03:A@37&B3:UC57~@=6< < C5::"7&>287"C87&>"2;B3B3:!, 180!NqM303:~4!:"Z3©N"287"C}@4!:7&25|%M3:0^]!M=:"TN TN&M=<37dNdZM32;03>":8:k<3;03:"TNSB32 v628B=`"S<=7"C}@q4!:ªpB3v~Z3q@^]UZ=<3":SB32c>:"_;0=N"C87&28714: n ,.1:" Z=:S28B=<327&:w3T3<>ªM30=N">":6 T:"B=321:"7&032;_8TN [0, n] C5::"7&>287"C87&>"v287ª>"[\<3;03[\SvM=2803>:"8:<3;03:"TN3, NC8C86:7&03<=6 T:"B32;w3B3v >°328M=`9N&03T:">UN k=1 k=1 00 n k=1 3 00 "[\<3;03[\ªNS4=Z3@{M32;03>":8:<3803:TNS>6:U628B379>"0=2;628B=< t 2;C87&` p .1:q1:0=6v~Z=21M3:&Z=B=:"D?>"280=:A@37&B3:C87&< ξt+1 = ξt + ηt , P {ηt = 1|ξt 6= 0 ∨ ξt 6= n} = p, t t t 6= 0 ∨ ξt 6= n} = q t kn (t) = P {ξt = n|ξ0 = k} P {ξt+1 = n} = <Z!< = P {ξ1 = k + 1|ξ0 = k}P {ξt+1 = n|ξ1 = k + 1}+ + P {ξ1 = k − 1|ξ0 = k}P {ξt+1 = n|ξ1 = k − 1} pkn (t + 1) = p pk+1,n (t) + q pk+1,n (t) k = 1, 2, . . . , n − 1 . , x P!, V a M 2;:"032~6v:9M303254!25Z!`"B3[¦|?>"280=:A@37&B3:C87~@|M303<=62;B3<37&`qB32k6:A©2;6¦]B3:q_~N"62;7&<=6¦]3w37": .1:&Z!:A©9µxzP=,JV"®M303< M N"TµTN&T ]=M3:&Z=vw3"2;03:A@37&B3:UC57&`9>"[\<=803[\N4=Z3@%M32;03>":8:q<3;03:"TN3]=7&: NC8C86N"7&03<3>UN&2~6Nd@{TN"T 1vB3T33<3@p:"7 k ]=>"280=:A@37&B3:C87&` p @3>&Z3@32;7"C}@0=28S2;B3<32~6tv0=N">"B32;B3<p@>9=T:"0,B=28w3B3p[¦|0==N&_;1B3:C87;@| x P=, pf −f +qf =0 v;4!:">"Z=287">":"0U@F>cS<=6?80!N&B3<3w=B3[6?vCZ!:">"<3@=6 f = 0 f = 1 , M 2;:"03<@k0=28S2;B3<@k7dN&T=<|Gv0=N">"B32;B3<3DkN"B=NdZ=:8<3w=B=N 7&2;:"03<3B328B=<3D?CGM3:UC57&:A@3B=B3[6<{TUN: =;F1<3=<328B37dN6<^, .1vUC57&`%C5B!NAw=NdZ!N ,28S2;B3<32qv842;6¨<=C5TN&7"`?>{><4!2 ]^}4!2 @3>"Z3@3287"C}@XT:"03B=2;6¡|N&0=N"T37&A2 @ 03<=C87&<3w32~C5T:";:%v0=N&>B328B3p<3@ 6=pλq − λ + q = 0 ,¬1:03B@=6"B3λ2;B3<@O@3>&λZ3F@ >c7"C}@ λ = 1, λ = q/p , )B=NAw=<37A]=1vB3T33<3< λ < λ v84:">&Z!287&>:"0UF@ > 7v0=N&>B328B3C< > xzP=, 5, F<3B328D=B=Nd@?TU:U6"<3B!N&3<@ x P!,Ja" f =C λ +C λ M303c"[¦| C < C 7AN"T©ª2ª@3>&Z=@3287"C}@µ032;S2;B3<32;6¦,^.1:~4)C57dN&>"Z3@@%80=N"B3<3w3B3[\2vC5Z=:"><@µ>Xx P=,a&5])M303< k = 0 < M3:&Z=vw3L4N[\<3803[\NM=2803>[6¡<380=:"T:6 7&:A©ª2v;4!:">"Z=287">":"0U@F>c7v0=N&>B328B= xzP=, 5,)b1:?80!N&B3<3w=B3[6< vC5Z=:"><@=6<{C57dN&B3v7 f = 1, f = 0 , E M=0324!28pZ3@@{<3_ =87&<|?vC5Z=:"><3D C < C ]=M3:&Z=vw39>"<42 n 1 n n pk0 = ((q/p) − (q/p) )/(1 − (q/p) ) . k0 kn 1 s M3:6:S`Q> 80!N&B3<3w=B3[¦|vUCZ=:>"<3DB=Nd|U:~4!<=6 2 fk = C1 + kC2 . pkn = 0=N">"B=N 2 ]7&:CG>"2;03:A@37&B3:UC57&`N> +G:~4<3B<=_F<3803:T:">>"[\<3;0=N&2;7~, .1vUpC57&`9+7&2;pM32803=` p1 = q = 1/2 , K =87":6 CZ=vw=N"2 λ = λ = 1 <{032;S28B=<32Fv0=N&>B328B3<3@Ox P!, U®B3vU©9B=:q<=C5TN&7"` k k , n pk0 = 1 − k . n K¨C}|2;621"Z=vU©ª4)N&B3<@?M3:9325Z![6t7&:w3TN"6 C1M3:" Z=:S2;B3<32;6t7&:&Z!`"T:ª>9B3v~Z=2F>2803:A@37"B3:C87&`qC5:"[\7&<@ >B=28T:"7&:03[\D%6:62;B379>"032~628B3< t, ξ ∈ [0, n] >:9>C52F6:U628B37&[ t} A = {ξ = 0 s®:"[\7&<@ n t t pk0 = An ( ((q/p)k − (q/p)n )/(1 − (q/p)n ), 1 − k/n, >&Z=:A©28B3[¨M3:C5Z=24!:>N"7&25Z=`B3:F403vF>S403v5N M3:N=87&:U6v>"2;03:A@37&B3:UC57&`9M3: Z=:"ª28B3<@>B=v;Z!2F0=N&>B=N 2~CZ=< 2;C5Z= p. 1< 8 % $&/ &%0:;: $p jUpr± ^ suZ=v w=N&D=B3:"DO1vUB=T33<32;DB=N"_8[\>N"287"C}@µC82;628D!C57&>:?C5Z=vw=N&D3B=[¦|{>25Z=<=w3<3B D ]_~N&>L< @ }C @3ª282S:7M!N&0=N6287&0=N t <3_96B3:A©2;C57">N T 2803:A@37"B3:C87&B3:6M=03ξ:C87&0=N&BF@ C87&>"2 U, P) ,#bFN&M3:U6B3<=6¦]w37":%M303UN&B3B3:U6 ω :7&:"0=(Ω, Nd©28B3<32 B=N"_8[\>N"287"C}@{>"[\:"03:w3B3:D?1vB3T3=<328D^,).103<µ1<3TC8<303:">UN&B3B=:6 _8B=NAw32;B3<3< t _8B!NAw328B3<=2 C5Z=vw=N&D3B=:"D?>"25Z!t<3→w3<3ξB3[ (ω)ξ (ω) B=N"_8[\>UN&287C}@>"[\:"03:w3B3[6/_;B=NAw328B=<32;6¦, O®CZ=< <<3B37"2803M30=287&<30=v"287C}@RTN&Tµ>032;6#@^])7&:?8:>":0@37:?C5Z=vw=N&D3B=:6M303:32;C;C52,)K=;7&:6 CZ=vw=N"2 _;B=NAw328B=<32"]M30=T<3B3⊂<=6RN"2;6:"2 ξ >6:U628B37q>032;62;B3< t B=N"_8[\>U?N > 7qC8:C57":A@3B3<32;6tM303:32;C;C8N3]>[\":03:"w3B=[\211vB3T=3<3< B=N"_8[\>U?N > 7?7&0=N"28T37":"03<@=6UNC5Z=vw=N&D3B=[¦|RM303:32;C;C8N{C57":A|UN"C87&<3w32~C5T3< =;T3>"<3>UNdZ=2;B37&B3[F]2;CZ!"M!NA4QN > 7SM=:"w37&<{B=N">"2;03B3:"2GM303c":U6 1<3TC8<303:">UN&B3B=:6 t ]7&:q2~C57&` P {ξ 6= η } = 0 , .1vUC57&` < ¯µCZ=vw=N"D3B3[\DX>"2;T37&:0^, OY8:{0=NC5M30=24!28Z=28B3<=2v;4!2;6 :"":_8B=NAw!N&7&` w32;0328_ F t(x, . ,. .. ., .t, x∈)T,N&. 7". `. , Tξ:B3)28w3B=:6280=B3[60=NC5M30=24!28Z=28B3<=2;6 CZ=vw=N"D3B3:"D{1vB3T33<=< ξ , :>":0@37A]w37&:ªC5Z=vw=N&D=B3[\DqM303:32;C;C_;Nd4N&B]"2;C5Z=<vTN&_~N&B3:FM30!N&>"<3Z=:14=Z3@:M303254!25Z=2;B3<@qT:"B32;w3B3:U62803B3[¦| 0=NC5M30=24!28Z=28B3<=D?C5Z=vw=N&D3B=:"D{1vUB=T33<3<4=Z3@?CZ >c":";:ªT:B328w3B=:"8:B=N"":"0!Nª6:U628B37&:>9B=N&"CZ >L4!28B=<@ t , . . . , t , E M3<=S2;6t7&03<µC8M3:C8:"NS_;Nd4N&B3<3@?CZ!vUw!N&D3B3:D?1vB3T=3<3< J +,¦W TN&_~N&7&`@3>"B=:M=0=N&><Z=:S4=Z3@{M3:C87&03:"2;B3<@?TU:B328w=B3:62;03B3[¦|0=NC5M30=24!28Z=28B3<=D I3,NA4N"287"C}@1vB3T33<@^]u_;N&><=C}@3ªN&@t:"7X>032;62;B3<=,YK TNAw=2;C57">"2GM=N&0=N6287&0=:">"280=vU7C}@CZ=vw=N"D3B3[\2G>"28Z=<3w3<3B=[¨Cc<3_8>2;C57"B3:"D?MZ=:7&B3:C87&Q` > 0=NC5M303254!25Z!28B3<@^,3bF?N @ M303<=62;0^]=M3vUC57&` g(t) = r sin(2πt + θ), t ∈ [0, 2π] <{M3vUC57&` r, θ C5Z=vw=N&D=B3[\21>"25Z!<3w3<3B3[ B=N9>2803:A@37&B=:C5?7 @ B3:6M303:UC57&0=N"B=C57">"2 (Ω, U, P) C%C5:>62;C87&B3:D MZ!:"7&B3:UC57&N` >E0=N"C8M303254!25Z=2;B3<@ F (u, v) , M :"}4N ξ (ω) = ¯>"[\:"03:w3B=N&@1vB3T33<@{CZ=vw=N"D3B3:8:M303:"=2;C8C;N ξ , t(ω) sin(2πt + θ(ω)), ω ∈ Ω H=,\¬1:"B32;w3B3:62;03B3[\210=NC5M303254!25Z!28B3<@?M3:"0=:A©ª4)?N > 7C}@M303<=62;B328B=<32;6 B328T:7&:"03:D:M3280!N&3<3"[\:"03:w3B3:"D 1vUB=T33<3"2;C87&B3:8:{M303:32;C;C8N ξη w328032;_9Z=<3B32;D3B3[\D/1<Z=`d7"0XC9C8>&@3_;N"B3[ <=B37&28;0=NdZ!:6tC8>"2;03T3< t t t t t t 1 t1 ,...,tn n 1 t1 t tn t n 1 n & r,θ t t t t ηt (ω) = Z∞ h(t − u)ξu (ω)du, u ∈ Ω. K4N&Z=`"B32;D3S2~6/Z=<=B328D3B=[\D{1<3Z=`d7&0?"v;4!28790=NC8C;6:"7&032;B{":&Z!282GM3:~4!03:"B3:!, ¬1:B328w=B3:62;03B3[\210=N"C8M303254!25Z=2;B3<@?v84:">&Z!287&>:"0UF@ > 7vC5Z=:"><@=6tC8:" Z!NC5:">UN&B3<3@ J +,-OuCZ="TN9w3<=C825Z 1, . . . , n < t , . . . , t ¯{M303:"<=_8>":"Z=`"B=[\2S6:62;B37&[B=N"&ZC>L428B3<3D] 7&:"}4N −∞ 1 I3,®QkZ3@ZC>c"[¦| n 1 n Fti1 ,...,tin (xi1 , . . . , xin ) = Ft1 ,...,tn (x1 , . . . , xn ); t1 , . . . , tn , tn+1 <3_ T Ft1 ,...,tn ,tn+1 (x1 , . . . , xn , +∞) = Ft1 ,...,tn (x1 , . . . , xn ) . M 28:"0=2;6Nq¥k, b9, ¬1:&Z!6:8:"0=:">NkvU7">"280U©4N"287 !#1' ± ) '&( ) ! # % ) , ! ) # , + , & + ! ) ) + ! )) # #'& '&,. #, '&.#'&'&.# ) ! !-,.! # + # # , t1 , . . . t n ∈ T '&") # % ' ' ) ! '&,. ' # ) ! # % # ! !-,..F! #t + ,...,t # #(x 1 ,! .,".! . , x n) )+(*R n / ) %-, ) ! # # "+ +' ,'&*% # % ' # # '&. {Ft ,...,t (x1 . . . , xn ), t1 , . . . , tn ∈ T, n = *") ,.., 1, 2, . . . } / 1 1 n n s®7dN&=<3:"B=N"03B3[6 M303:32;C;C5:U6¨B!N&_8[\>UN&2;7"C}@XC5Z=vw=N&D=B3[\DM303:32;C;C&]pTU:B328w=B3:62;03B3[\20=NC5M30=24!28Z=28B3<3@XT:Q@ 7&:03:";:B=2G_~N&><=C}@37ª:79B=NAw=NdZNS:7"C5w32;7dNª>032;62;B3< Ft1 +t,t2 +t,...,tn +t (x1 , x2 , . . . , xn ) = Ft1 ,t2 ,...,tn (x1 , x2 , . . . , xn ) . QkZ3@¡4<=C5T30=287&B3[¦|¨M303:"=2;C8C8:"> x 6B3:A©ª2~C57&>: C8:C57":A@3B3<3D T:7&:"03:8:B=2R:&Z=2;2Rw32~6 C5w=287&B3:C87dN&3<3:B=N&0=B3:C87&` :"_;B=NAw=N"287 P {ξt1 +t = x1 , . . . , ξtn +t = xn } = P {ξt1 = x1 , . . . , ξtn = xn }. QkZ3@{N&C8:&ZC>c7&B3:9B32;M303280=[\>"B3:8:9CZ=vw=N"D3B3:";:ªM=03:"32~C8C;NC87dN&3<3:B=N&0=B3:C87&`ª:_8B=NAw!N&287A]w37": pt1 +t,...,tn +t (u1 , . . . , un ) = pt1 ,...,tn (u1 , . . . , un ) . E 7"62;7&<=6¦]=w=7&:6:62;B37&[°C57dN&3<=:"B=N"03B3[¦|M30=:"32~C8C5:U6<3B=>N&0=<=N&B37"B3[ M3:9>032;62;B3<^,QkZ3@?M303<=62;0=N n M (ξt1 ) = Z∞ n x pt1 (x)dx = OuC5Z=< ¯X1vUB=T33<@ M3280=2;628B=B3[¦| < ©4!28B=B=gNd@%C57dN&=<3:"B=N"03nB3[6 CZ=vw=N"D3B3[6x M=,03. :". .3,2~xC8C8:6 gξ ] 7&: −∞ 1 n Z∞ xn pt2 (x)dx = M (ξt2 )n −∞ = g(ξt1 , . . . , ξtn ) t1 ,...,tn ¯OC5Z=vw=N&D3B!Nd@/>"28Z=<3w3<3B!N3]M=:"03:Q@ t M gt1 ,...,tn = Z g(u1 , . . . , un )pt1 ,...,tn (u1 , . . . , un )du = Rn = Z g(u1 , . . . , un )pt1 +t,...,tn +t (u1 , . . . , un )du = M gt1 +t,...,tn +t . ¯?M303:<3_8>:&Z=`B=Nd@?M3280=<3:~4!<3w32~C5TNd@µ1vB3T3=<@CFM32;03<3:~4!:U6 , E M303254!25Z!<=6 CZ!vN@ ! # ! .1vUC57&` w=N"D3B3[\DµM30=:"32~C8C / ξ ]M=:&Z=:Ag(t) ©q<3> ξ = g(t + τ ) ]}42 τ ¯%CZ=vw=N"D3B=Nd@%>25Z=<3w=<3B=N3]0=N">"B3:U62803TB3:0=N"C8M303254!25Z=2;B3B=N&@ B=N<3B37"2803>UNdZ=2G4=Z!<3B3:"D T , M :"}4N ξ ¯C57dN&3<=:"B=N"03B3[\D{CZ=vw=N"D3B3[\D?M303:"=2;C8C", suZ=vw=N"D3B3[\DXM303:32;C;C CB32;M303280=[\>"B3[6¨>"032~628B32~6 B!N&_8[\>UN&2;7"C}@M303:32;C;C5:6¨CB328_~N&>" "[¦| t , . . . , t , 0 6 t "28Z=<3w3<3B=[ Rn t t t t 1 n 1 n ξt1 , ξt2 − ξt1 , . . . , ξtn − ξtn−1 3B 2;_;N&><=C5C8T3<=6RM303:"=2;C8C8:6XB=N"_8[\>UN&287C}@9M=03:"32~C8C ξ C5:ªC8w3287"B3[6/w3<=C5Z=:6tC5:C87&:A@3B3<3D]v;4!:>&Z=2;7&>":0@F>9@ S<=D?vUCZ=:>"<@=6 ¯M303:32;C;C1C1B328_~N&>" "[¦| t < t < s M30=<30=N&ª28B3<3@ ξ −ξ < ξ −ξ :~4!<=B=N&T:">:\0=N"C8M303254!25Z=2;B3[ xz:~4!B3:"0=:~4!B3:C87&` 2 . M3:9>"032~628B3 2} = o(h), P {ξh = 1} = λh + o(h), 0 < λ < ∞ , 7&:ª2~C57&`>"2;03:A@37&B3:UC57&`C5TNAw3TNk>B3v7&03<TU:03:"7"TU:8:ª<3B37&2;03>N&Z!Nk>032;62;B3<M=03:~4!:&Z=©9<=7&25Z=`B3:C87&`Q> h 0=N&>B=N ]!N>2803:A@37&B=:C57"`ªB=N&Z=<3w3<3@?:&Z=2;2c:~4!B=:"8:9C5TNAw3TNª0=N&>B=N o(h) , λh + o(h) +~g"g !#1'µ±p-¢ ξt ) ,. . ! .! # #2'& P {ξt = k} = (λt)k −λt e . k! ¯/M303:<3_8>:~4=@3ªN&@/1vB3T=3<@ C5Z=vw=N&D3B=:"Dt>25Z=<=w3<3B3[ ξ , M N&T TN&T < ξ ¯qB=28/ _;N">"<=C8<=6ϕ[\2F(x)CZ=v=w=N"MD3B3x[\21>"28Z=<3w3<=B3[F]7&: *! , ,' # ( '& .1vC57"` ξt+h − ξt t ξt t t ϕt+h (x) = M xξt+h −ξt M xξt = M xξh M xξt . .1::"M30=24!28Z=28B3 M3vNC5:"B=:">C8T:"8:SM303:"=2;C8C;N M xξh (x) = 1 − λh + xλh + o(h), .1:N=87&:U6v ϕt+h (x) = ϕt (x)(1 − λh + xλh + o(h)) . E 7"CA>L4N9CZ=254!v"2;7 K¡M=0324!28Z=2FM3:"Z=vw=N&2~6 |x| 6 1 . ϕt+h (x) − ϕt (x) = λ(x − 1)ϕt (x) + o(1) h M=03< h → 0. dϕt (x) = λ(x − 1)ϕt (x) . dt 28ª28B3<=2;6 vU0!N&>"B=28B3<@@3>&Z3@32;7"C}@?1vB3T3=<@ eλt(x−1) = ∞ X (λt)k k=0 k! e−λt xk . , #%-, # .1vNC8C5:B3:">UC5T3<=DtM303:32;C;Cµ6:A©qB3:/_;Nd4N&7"`/B=N/>"2;03:A@37&B3:UC57&B=:6M303:C87&0=N"B=C57&>2 21C5TU,NAw3PT3<) ]] }4!2 Ω C5:>"M=Nd4N&2;7SCG/ 6B3:A©ª2~C57&>:6/C57"vUM=28B3w=N"7&[¦|1vB3T3=<3D^]vqT:"7&:03[¦|q<=62A>c7"C}@7&:"Z=`"T:ª24!<3B=<3w3B3[\(Ω, <{6:62;B37&[°>"032~628B3<]3C5:":7&>"2;7"C57">"v > ª<32FC5TNAw3TN"6¦]B321<=62"> 7q7&:w328T{B=N"T:"MZ=2;B3<@^, s®: Z!N"C8B3:k7&2;:"032~62 ξ M303<CZ >c":6t1<3TC5<303:>N"B3B3:6 t 0=N&>B3:Sw3S7&0!N&28T=7&:"03<=<M303:32;C;C8N ξ B=N:7&0328_;T2 [0, t] , s°M3:6:SQ` > 4!:TN"_;N"B3B3:"D7&28:032;6[ 6:A©qB3:?B=N"D37&U62;C87&B3:"2ª0=N"C8M303254!25Z=2;B3<32qCZ!vUw!N&D3B3[¦|µ>25Z=<=w3<3B 4!Z3@KZ > "[¦| 0 6 t < · · · < t , 11C8B3:=]w=7&: ξ ,...,ξ t t1 tn s 1 n {ξt1 = k1 , ξt2 = k2 , . . . , ξtn = kn } = {ξt1 = k1 , ξt2 − ξt1 = k2 − k1 , . . . , ξtn − ξtn−1 = kn − kn−1 } . 1_1B32;_;N&><=C5UC5T3<=DM303:32;C;Cc:&Z!Nd4N&2;7ªC8>":"D!C57&>:6t6N"03T:">:C57"<^, SQ :UC57dN&7":"w3B3:\/ >:C8M3:&Z=`_8:>N&7"`C}@ :M3032425Z=2;B3<32;6?vUCZ=:>"B3:DG>2803:A@37"B3:C87&":D=C57">":6N&0F@ T:">:C87&<M3vUN"C;C5:"B=:">C8T:"8:SM303:"=2;C8C;N3, sL1:0=6v~Z=<303v2;6t_4!2~C5`97&28:032;6v]3T:"7&:0=Nd@?M3:"7"0328v"287C}@>ª4NdZ=`B328D=S2;6¦, +;g=+ =1'± ) ' ( ) %-, , '& #!$# ! . ...,ξ ) .#! '& ! &#!$# ! . ' η' =( (η !-,,.....! , #η+# ) #= g(ξp ,(x) # x = (x , . . . , x ) g(x) = (g (x), . . . , g (x)) / '& !-, # # ., "+.,"% # #.! # ! + # !$# ! ) # & ! , y = g(x) 1 r 1 r ξ J(x) = 1 r 1 r ∂(g1 , . . . , gr ) 6= 0 , ∂(x1 , . . . , xn ) '& !-,.! # + # # # .# ! ' !-, η ,. '& #!$# ! .& pη (x) = pξ (g −1 (x))|J(g −1 (x))|−1 . .1vUC57&` ¯6:62;B37 @ 8:ªC5TNAw3TN17&0=N"28T37&:03<3<< τ = θ − θ ¯ M303:628©qvU7":"T>032;62;B3< 625©4!v4!>"v6#@?θ M3,:kC5Z==24!0,:>1,N"7&.25.Z=. `B3[6<{C5TNAw3kTN"6<^, =1'S±p ! n )%-, # .# % τ , . . . , τ # ,. ! , + , ##' k k k k−1 *! ,.,.'&# ( # !-,..! # +# # # , !-, #' ! λ / *! , ,' # ( ' suZ!vUw!N&D3B3[\2G>"28Z=<3w3<=B3[ τ =< 62A>c79M3:"TN&_~N&7&28Z=`"B=:"20=NC5M30=24!28Z=28B3<=2GCGM=N"0=N"62;7&03:6 λ ,11CR@ / 1 B3:!]w37&: QkZ3@ n k P {τ1 > t} = P {ξt − ξ0 = 0} = P {ξt = 0} = e−λt . k>1 M3:1:"0=6v~Z=2GM3:&Z!B3:"D>2803:A@37"B3:C87&<<C8>":D=C57&>v9:~4!B3:"0=:~4!B3:C87&"032~628B3<M303:32;C;C8N ξ B=Nd|U:~4!<=6 t P {τk > t} = Z P {ξt+θk−1 − ξθk−1 = 0|θk−1 = u}pθk−1 (u)du = R = P {ξt = 0} 1. vUC57&` < M30=:625©qv7&T3< M32;032;C828TN?>c7"C}@^0,3.1=:&Z=t:A©q<<=6 th <= ·0·, ·M 0 k = 1, . . . , n n ]LB32 M3:N=87&:U6v {θ1 ∈ ∆1 , θ2 ∈ ∆2 , . . . , θn ∈ ∆n } = P{ n \ k=1 (ξk ∈ ∆k )} = =e }4!2 .1vUC57&` ∆ = ∆1 × · · · × ∆ n sk = gk (u1 E 7&:"0=N&©ª28B=<32 < n Y n \ k=1 {(ξtk − ξtk−1 +hk−1 = 0) k=1 λ e h1 . . . h n = Z ··· Z pθ1 ...θn (t1 , . . . , tn )dt1 . . . dtn + o(1) , ]32;C5Z=< 0 < t (t , . . . , t ) = e λ p ] <{M3vC87&` ...u ) = u − u k = 1, . . . , n n (ξtk +hk − ξtk = 1)} , (e−λ(tk −tk−1 −hk−1 ) λhk e−λhk ) −λtn n −λhn θ1 ...θn \ 1 n k ∆ −λtn n k−1 1 < · · · < tn , (τ1 , . . . , τn ) = (g1 (θ1 , . . . , θn ), . . . , gn (θ1 , . . . , θn )) ] u = (u , . . . , u ) >_;N"<=6B3::~4B3:"_;B=NAw3B3:<@3T:"""28BO+",Wcw3<37&[\g(u) >UNd@^]3=w=7&(g: u(u),=. .s. , +g ·(u)) ]=<3_1M30324[L4!vS28D{7&2;:"032~6[¨M=:&Z=vw=N"2;6 ·· +s 1 n n 1 1 n n pτ1 ,...,τn = λn e−λ(s1 +···+sn ) = }4!2 n Y pτk (sk ) = λe−λsk pτk (sk ) , ¯MZ=:7&B3:UC57&`9M3:TN"_;N"7&25Z=`B3:";:S0=NC5M30=24!28Z=28B3<3@{C1M=N"0=N"62;7&03:U6 λ , k=1 +~gI !²#" r "jrr¡#'G#R#r# N&03T:">UC5T3<=DµM303:32;C;C ξ B!N&_8[\>UN&2;7"C}@µM303:32;C;C5:6;<3"28Z=<[\M3:&ZC@ B@F>c7"C}@{vCZ!:">"<3@ +, P {ξ = k + 1|ξ = k} = λ h + o(h) I3, P {ξ = k − 1|ξ = k} = µ h + o(h) H=, P {ξ = k|ξ = k} = 1 − (λ + µ )h + o(h) ] }4!2 λ , µ > 0 M303< k > 0 ]3< µ = 0 M303< k 6 0 , G12;0328_ :":"_8B!NAw3<=6{>"2;03:A@37&B3:UC57&`c7&:";:=]Aw37&:1>16:62;B37 t M303:"=2;C8C®B=N&|:~4!<37C}@S>GC8:C87&:A@3B3<3< k , M :"}4)N M3:q1:0=6v~Z=21M3P:&Z=(t)B=:"D?>"280=:A@37&B3:C87&< t t+h t k t+h t k t+h k t k k & & k k k Pk (t + h) = Pk (t)(1 − (λk + µk )h) + Pk−1 (t)λk−1 h + Pk+1 (t)µk+1 h + o(h) . WYC87&032;6#Z=@@ h T{B3v~ZC>ª]3>9M303254!25Z=2FM=:&Z=vw3<=6 Pk0 (t) = −(λk + µk )Pk (t) + λk−1 Pk−1 (t) + µk+1 Pk+1 (t) < P00 (t) = −λ0 P0 (t) + µ1 P1 (t) . OuC5Z=<_~NA4N"B3:%B=NAw=N&Z=`"B3:290=N"C8M3032425Z=2;B3<32 {P (0)} ] P P (0) = 1 ]p7&:RC8<=C57&2~6N?4!<=F12;0328B=3<=NdZ!`"B3[¦| v0=N">"B32;B3<3D°<=6282;7¡24!<=B=C57&>28B3B=:"2032;S2;B3<32"]c2;C5Z=< T:Q=;F1<=3<328B=7&[ λ < µ :";0=N&B3<=w328B3[ <Z!< >":"_;0=N"C87dN @ >c7/4!:UC57dN&7&:w3B3:t6254=Z=2;B3B3:=,\K8= 7":6 CZ!vUw!N&20=28S2;B3<32 {P (t} v84:">&Z!287&>:"0U@3287/vUPCZ=:>" 0=288v~Z3@30=B3:C87&"[\0=N&7"`µT:Q;= F1<=3<328B=7&[ 7dN&T3:">UN&7&`X"2~C5T:"B=28w3B3:2%6B3:A©ª2~C57&>:O0=28S2;B3<3D^,¦bFNXC5v66v 6:A©qB3:tC;6:"7&0=287&`TN"T B!N >2803:A@37&B=:C57"`7&:8:=]w37":R_~N>032;6#@ t M303:<3_8:D4!287Z=<3ª`OT:B328w3B=:"2w=<=CZ=:O<3P_~6(t)28B32;B3<3DC5:C87&:A@3B3<3D C5<=C87&2;6[F, KL280=:A@37&B3:C87&` P >µ7dN&T:6 CZ=vw=N"2q<3B37&2;03M3032;7&<303v287"C}@TN"TO>2803:A@37"B3:C87&`{"2;C8T:"B32;w3B3:";:{w3<=CZN <3_~628B32;B3<3D C5:1C87&−:A@3B3<@C8<=PC57&(t)2~6[ _;N>"032~6#@ t ,#KyvUCZ=:>"<@|X0=NA4<3:N"T37&<3>B3:";:%0=N"C8M=NA4Nµ7AN"TNd@>":_;6:A©qB3:UC57&` :_8B=NAw=N"287SZ!N&><3B3B3[\D?0=N"C8M=NA4#, =²"?mpr&3 X' G Upr# .1vUC57&` µ = 0 < λ > 0 ,!s®<=C87&2;6Nª4<=F128032;B33<=N&Z=`"B=[¦|vU0!N&>"B=28B3<3D?D> =87&:U6 C5Z=vw=N&21M30=<=6287q><4 ∞ k=1 k ∞ k=1 ∞ k=1 ∞ k=1 k k k k k x P!, U P00 (t) = λP0 (t) Pk0 (t) = −λPk (t) + λk−1 Pk−1 (t), k > 1. .1:UCZ=254!:">UN&7&28Z=`"B3:S<3B37&2;803<30=v&@^]=B!Nd|:~4<=6/>C82 P (t) , !#1'± * x S28Z3Z=280, ' ") #% '& ! .# t !$#% # k # ) + #'& ! ) ! # ) ) ! '& ! + !-,. "+ / ∞ X k=1 P∞ k=1 λ−1 k )+!-,.# % '&") &! , P (t) # "+ + '&,.' *% #% '& P (t) = 1 k k ;+ gH *! , ,' # ( ' .1vUC57&` S (t) = P (t) + . . . P (t) ,s®T3Z!NA4![\>UNd@M=2803>[\2 n v0=N&>B328B=<3D>%xzP=, ]=M3:&Z=vw=N"2;6 / n n Sn0 (t) = −λn Pn (t). .1:Q=;7&:6v M N&T%TN&T Sn (0) % 1 < Sn (0) − Sn (t) = λn M30=< Sn (t) % n→∞ Zt Pn (t)dt. ]=C5vS2~C57&>v"2;7ªM30=24!28Z lim [Sn (0) − Sn (t)] = µ(t) > 0 . ¥°_8B!NAw3<37A]4=Z3@%M3:"Z=:A©q<37&28Z=`"B3:8: ε M303<µ1<=TC8<303:>N&B=B3:6 t ]3B!NAw3<3B=N&@?C1B32;TU:7&:"0=:"8: n ]=>"[\M=:&Z=B@32;7"C}@B=280=N?@ >"2;B=C57">": n→∞ λn .1:Q=;7&:6vq4=Z3@4!:C87AN"7&:"w=B3:ª:&Z=`S<| n <=62;2;6 t> Zt Zt Pn (t)dt > µ(t) − ε. Sn (z)dz > (µ(t) − ε) 1 1 1 + +···+ . λ0 λ1 λn 1_ ;= 7&:";:B32;0=N&>28B=C87&>NRCZ=254!v"2;7A]pw37":R2~CZ=< 0@40=N"C}|U:~4!<37"C}@^]p7&: M303:<3_8>:&Z=`B3:=]7&: µ(t) 6 0 ,3.1:N=87&:U6v s4!03v8:D?C87&:"0=:"B3[F]@=C8B3:=]3w=7&: K¡M30=24!28Z=2FM303< OuC5Z=< M3:&Z!vUw!N&2;6 n→∞ ]7&:92;C87&` µ(t) = 0 ]3N_8B!NAw3<37A] S (t) %µ(t) , −ε 6 0 µ(t) = 0 1 R ,3.1N: =87&:U6v λ P (t)dt 6 1 Zt t n 0 ε n n Sn (t)dt 6 Zt M N"T TN"TM3:&Z!:A©9<=7&25Z=`B3:"2 1 1 1 . + +··· + λ0 λ1 λn [1 − µ(t)]dt 6 ∞ X 1 . λ n=1 n ] 7&:q0U@4%C8M30=N">N0=N"C}|U:~4!<37"C}@^, iSr#"3U1'O1'3"=() / djl Xrp()'#rp{$#!3U^Gr .1vUC57&`F7&0=28":>N"B3<@^]&M3:UC57&vM=NQ> ª<32\B=N :"UCZ=vU©q<3>QN > ª<|SvC87&03:D=C57&>F:"0=N"_8v > 71M3vUN"C;C5:"B=:">C8T3<3DM303:Q@ 32~C8C9CSM!N&0=N6287&03:U6 λ ]N>"032~6#@R:"UCZ=vU©q<3>UN&B3n<@%KZ > ":8:?7&032;":">UN&B3<3@%CZ > [6 vC87&03:D=C57&>:6¡"NQ> S<3|µvC57"03:"D=C87&>=, OuC5Z=<>C82:"UCZ=vU©q<3>UN?> ª<32\vC87&03:D=C57&>UNF_;N"B@37&[F]U7&:S7"0328:">N"B3<32\7&2;0U@3287"C}@^, G1<=CZ!:k7&032;":">UN&B3<=Dq>SC5<=C87&2;62 ξ >q6:62;B37 t 6:A©2879M303<3B=<=6N&7&`q_;B=NAw32;B3<@ 1, . . . , n , b12;7&03v;4!B3: M3:"B@37"`=]uw=7&:>"2;03:A@37&B3:UC57&`/M=280328|:~4NX<3_RC5:C87&:A@3B3<@ k >C8:C57":A@3B3<32 k _;N>"032~6#@ h → 0 :"7~Z=<=w=N&2;7"C}@B=N o(h) :7ª>2803:A@37&B=:C57"<M30=:"<3_;>"2428B3<@{C5:"[\7&<3D _;N>"032~6#@ h B321_~N&T:"B3w=<37"C}@:C5Z=vU©9<=>N&B=<32cB=N&B=<3D } < { _;N>"032~6#@ h B321M3:UC57&vM3<=7qB3:">[¦|7&0328:">UN&B3<3D } , { limn→∞ Sn (t) = 1 t +~g&P M N&T{TN&T?vC57"03:"D=C87&>N:CZ!v©q<3>UN?>c7"C}@B32;_;N&><=C5C8:C87&:A@3B3<32 k + 1 _~Nq>"032~6#@ h → 0 :7~Z=<3w=N"287"C}@%B=N >2803:A@37&B=:C57"<M30=:"<3_;>"2428B3<@{C5:"[\7&<3D _;Nª>"032~6#@ h B32F_~N&T:"B=w3<37"C}@?:"C5Z=vU©q<3>N"B3<32GB3<{:~4!B3:";:ª<=_ k 7&0=28":>N"B3<3D } < { _;Nª>"032~6#@ M3:UC57&vM3<=7q:~4!B3:B3:">:"2G7&0328:">UN&B3<32 } , { K =87&:U6 C5Z=vhw=N&2 P {ξt+h = k + 1 | ξt = k} = λh + o(h) k < n. KL2;03:A@37&B3:UC57&`qM32;0325|U:~4N9<3_C5:C87&:A@3B3<@ >C8:C87&:A@3B3<32 _~Nq>"032~6#@ >2803:A@37&B=:C57"<_~N&T:"B3w=<37&`:"C5Z=vU©q<3>N"B3<32c|:"7~@?k"[ :~4B3:";:ª7&0=28k"−:>1N"B3<@^, P {ξt+h = k − 1 | ξt = k} = kµh + o(h) , h→0 :7~Z=<3w=N"287"C}@%B=N o(h) :"7 o(h) :"7 us Z=vw=N"< k = 0 < k = n 0=N"C;C86N&7"03<3>NQ> 7"C}@?:"7;4!28Z=`"B3:!, s®":";:9:"C5Z=vU©q<3>N"B3<@%CkM3:7&280U@=6<µ>M3<=C5[\>UN&2;7"C}@?>M303:"=2;C8CS8<325Z=<%<0=N&_~6B3:A©ª2;B3<@µC M=N"0=N"62;7&0=N6< λ = λ (k = 0, 1, . . . , n − 1) ] λ = 0 (k > n) ] µ = kµ (1 6 k 6 n) ] µ = 0 (k > n, k = 0) , KM3032425Z=29M303< h → 0 M303<|U:~4!<=6 TOC8<=C57&2~629Z=<3B32;D3B3[¦|v0=N">"B32;B3<3Dµ4=Z3@XB=Nd|U:A©ª428B3<@µ>2803:A@37"B3:C87&< 3 M 0=:"32~C8C5vB=Nd|U:~4!<37&`UC}@>qC5:C87&:A@3B3<3< k >6:628B=7>032;62;B3< t , P (t) k k 0 < k < n. k k k P00 (t) = −λP0 (t) + µP1 (t) Pk0 (t) = −(λ + kµ)Pk (t) + λPk−1 (t) + (k + 1)µPk+1 (t) , Pn0 (t) n X 0 < k < n, = −(λ + nµ)Pn (t) + λPn−1 (t) , Pk (t) = 1 . QkZ=@ vUC57dN&B3:>"<3>S2;8:C}@ M303:"=2;C8C;NO>">"254!2;6M303254!25Z!`"B3[\2µ>2803:A@37"B3:C87&< B=N&|:~4=@37C}@<3_FC5<=C87&2;6[°v0=N">"B32;B3<3D k=1 pk = limt→∞ Pk (t) λp0 + µp1 = 0 , E :"_8B!NAw3<=6/w32;0328_ E 7"CA>L4NB=Nd|U:~4!<=6 λpk−1 − (λ + kµ)pk + (k + 1)µpk+1 = 0, λpn−1 − nµpn = 0 n X pk = 1 . k=0 zk = λpk−1 − kµpk ]=v84:">&Z!287&>:"0U@F> ª<32GC8<=C57"2;62 z0 = 0 , zk − zk−1 = 0 , 1 6 k < n, zn = 0 . kµpk = λpk−1 , k = 1, . . . , n. 1 6 k < n, ]®TU:7&:"0=[\2 +;gUV .1:Q=;7&:6v pk = M N"T{TN"T ρk p0 , k! p0 B=N&|:~4!<=6 k > 1, n X ρk k=0 p0 = 7&2~6 C;N"6[6¦]4!:TN&_;N"B=N1:"0=6v~Z!N£®0Z!N"B35N k! = 1, n X ρk −1 k! k=1 λ . µ ρ= ρk X ρk −1 , k! k! , n pk = 1 6 k 6 n. .103< M3:"Z=vw=N&2~6%>"2;03:A@37&B3:UC57&`c7&:";:=]Aw37&:G>C82uZ=<=B3<3<_;N&B3@37&[F]&N _;B=NAw3<37A]A>"B=:">"`cM3:C87&vM3<3>S282®7&032;":">UN&B3<=2 "v;4!2;79k :"=7&>n2803;B3v7&:=]7&:q2~C57&`9>2803:A@37&B=:C57"`ªM3:"Z=vw3<37&`q:7&TN&_ k=1 ρn X ρk −1 . n! k! n pn = 3 'l!"!("pr#n &j#lm'n#po9n $#=²" k=1 @3>&Z3@F>c7"C}@ $p j#r ±)¢ OuC5Z=<T:B328w3B=[\2k0=N"C8M303254!25Z=2;B3<@%M303:32;C;C8N 4!Z3@%ZC> [¦| B3:0=6NdZ=`B3:90=N"C8M303254!25Z=2;B3B3[6C8T3<=6N"28ξ7"C}@?5NAvC;C5:">UC5T3 7"C}@>28T37":"03:U6 M (ξ , . . . , ξ ) = (M ξ , . . . , M ξ ) <T:">UN&03<=N"3<3:B3B3:"Dµ6N"7&0396:628B=7&[¨>032;62;B3< $p j#r ± ^ QkZ=@_;Nd4N&B3<3@T:">N"03<=N"3<3:"B=B3:"D6N"7&03<3=[ B32;:"A|:~4<=6:c<4:C57dN"7&:"w3B=:F_8B=N"7&`k1vB3T33F< > B(t, s) ]"B=N"_8[\>UN&2;6v >¡T:">UN&03{bqB!N&&ZK>L4!28B3<3@ t < Bξ (t, s) = M (ξt − mξ (t))(ξs − mξ (s)) . i tj bij = Bξ (ti , tj ) . NAvUC8C5:>C8T3<3DkM30=:"32~C8CL_;Nd4N&B^]d2~CZ=<<3_;>"2;C87&B3:1TN&T3N"03<=N"3<3:"B=B=Nd@1vB3T3=<@ B(t, s) = M (ξ − m(t))(ξ − m(s) _~N&><=C}@37ª:76:U628B37":">qmB=N"(t)&ZC>L=4!m(t) 2;B3<@^=, M ξ $p j#r ± ¬1:"030328Z3@33<3:B3B=N&@1vB3T33<@ R (t, s) M30=:"32~C8C8N ξ :"M=0324!28Z3@3287C}@0=N">"28B!C57&>:6 ξ t t s ξ SvB3T33<3< B3<| "@3_;N&B=[ C5::"7&B3:S2;B3<32;6 R (t, s) ξ E 7"CA>L4N C5Z=24v"287A]~w37&: 4=Z3@k:M3032425Z=2;B3<@k5NAvC;C5:>C5T:8:LM303:"=2;C8C;N ξ 4!:UC57dN&7":"w3B3: _8B=N"7&`c1vB3T3= C80324C@ Bξ (t, s) = M (ξti − M ξti )(ξtj − M ξtj ) = M (ξt ξs ) − mξ (t)mξ (s) = Rξ (t, s) − mξ (t)mξ (s) . ?< Z=<3": ]Z=<=": , ¬1:030325Z=@33<3:"BB=B=(t,Nd@qs)1vB3T33<@qRC(t,Z!vUw!s)N&D3B3:8:1M303:32;C;C8N ξ B32;:"7&03<==N&7&28Z=`"B=:k:"M30=24!28Z=28B=N=,AQS28D!C57&><37&25Z!`"B3:!] :k"[ B3<[¦Z="<37"25Z=`B3[\2cw=<=CZ!N t , . . . , t ]T:6MZ!28TC5B3[\2 c , . . . , c <=25Z=:2 R(t, s) = R(s, t) w3<=C5Z=: n cS282FC8:":"7"B3:"ª28B3<32 t mξ (t) ξ ξ t 0 6 M| n X k=1 c k ξ t k |2 = M { n X n X i=1 j=1 1 n c i c j ξ ti ξ tj } = n X n X j=1 i=1 1 R(ti , tj )ci cj . n +~g" QkZ=@ 5NAvC;C5:>C5T:8:%M303:32;C;C8NCZ > UNd@ MZ=:"7"B3:C87&`/C5:">U62;C87&B3:";:R0!N"C5M=0324!28Z=28B=<@ >[\":03T3< ~_ N&>"N"03<=N"3<3D b ,.1:Q=;7&:6v_8B=N"B3<32 m (t) 5T:";:SCZ=vw=N"D3B3:8:M303:"=2;C8C;N3, SvB3T3=<3< ] ] 6:"8v7[\7&`7&:w3B3: 7AN"T©ª2:"M303254!25Z!28B3[E<4!Z3@ B3285NAvUC8C8:">C8T:"8: C5Z=vw=N&D3B=:"8:M303:"=m2;C8C;(t)N ξ B,K°(t,B32;s)5NAvRC8C8:"(t,>UC5s)T:6 CZ=vw=N"2S_8B!N&B3<32ª7&:&Z!`"T: =;7&<|1vB3T33<=D^])>":":"ª2F8:">:"0U@^]=B=24!:Q@ C87AN"7&:"w=B3:ª4=Z3@7":"8:!]3w37&:"[¨M3:&Z=B=:C57"Q` >°_;Nd4N&7&`9M30=:"32~C8C&, QkZ=@¨CZ >c":D 4!28D=C87&>"<=7&25Z=`B3:"D 1vB3T3=<3< B=N <°CZ > :"D¨4!2;D=C57&><37&28Z=`"B3:D B328:7&03<3!N&7&28Z=`"B3:D :M3032425Z=2;B3B3:"D¨1vUB=T33<3< R(t, s) B!N T × T C5vSm(t) 2~C57&>v"2;7T5NAvC;C5:">UC5TNd@¡CZ!vUw!N&D3B=N&@ 1vB3T33<@ ξ C m(t) = < s) > TNAw32~C57&>2T:"0=0325Z3@3=<3:"B3B=:"D 1vB3T3=<3<^,\K TNAw32;C87&>"2 F "2;032;6 @ 62;03B3:"2OB3:0=6NdZ=Q` @ Mξ B3:2k0=NC5M3R(t, 03254!25Z!28B3<32Ck>28T37&:03:6¡6N"7&2;6N"7&<3w32~C5T3<3|{:A©q<4)N&B3<3D (m(t ), . . . , m(t )) <6N&7"n03<332;DT:">UN&03<=N"3<3D ,=QkZ3@C5vS2~C57&>:">UN&B3<@?7dN&T:">:"8:4:C57dN"7&:"w3B=:B328:7&03<3!N&7&28Z=`"B3:D%:"M30=24!28Z=28B3B=:C57"<6N&7&0=<33[ {B(t , t )} T:">UN&03UN&B3B3:UC57&`S0=N"C8M3032425Z=2;B3<3D?>"[\7&2;TN"287<3_F7&:8:=]w=7&:qB3:"0!6NdZ=`B3:"2G0=N"C8M303254!25Z=2;B3<321:~4!B3Q: @ _;B=NAw3B3::M303254!25Z3@32;7"C}@µ6N&7&2~6N&7&<=w32;C8T3<=6 :A©q<4)N&B3<32~6 <6N&7&0=<3328DRT:">N"03<=N"3<3D%<µ7&2~6¦])w37&:0!N"C5M=0324!28Z=28B=<32 CZ >c":8:M3:~4!>"2;T37&:0=NSB=:"0=6N&Z=`"B3:8:>"2;T37&:"0!NC8B3:">UNSB=:"0=6N&Z=`"B3:%x C86¦,37&2;:"032~6v¬1:&Z6:"8:03:">UN5, ! # ! .1:{7&2;:"032~672 S®:A|B32;0=N @ <=B3w3<3B=N?1vB3T33<3@ exp(−|u|) @3>"Z3@3287"C}@OT:"030328Z3@33<3:B3B3:DR1vB3LT @ 3<=28D{5NAvC;C5:">UC5T:";/ :kCZ=vw=N"D3B3:8:M303:"=2;C8C;N3, NAvUC8C8:">C8T3<3DªM303:"=2;C8C @3>"Z3@3287"C}@qC57dN"3<3:"B!N&03B3[6¦]"2;CZ!<3821B=NAw!ξNdZ!Nª:"7"C8w3287dN=,3.1Q: =87":6v94=Z=@{C87AN"3<3:B=N&03B=:"8:M303:32;C;C8N m (t) = M ξ = M ξ = m (t + τ ) = m = < − m = B (t + τ, s + τ ) B (t, s) = M ξ ξ − m = M ξ ξ 4=Z=@k>C528| t, s ∈ T ,;KXw!N"C57"B3:C87&<^]~2;CZ!C528| t < s 4!:&Z=©9B=: >"[\M=:&Z=B@37"`C}@F0=N&>28B=C87&>": ]®7&:2~C57&`/T:">UN&03"<=C8287&`X7&:"Z=`"T:/:"7 4=Bξ(t, Z!<3B3[/s)<3B3=7"28Bξ(t 03>UNdZ!N−6s,28©0)4!vG=6:UBξ(t 628B37d−N6s)L4!2;B3<@^,8WYC5Z=:"><@^]8M=03UC5T3<3DGM303:32;C;C@3>&Z=@3287"C}@ C87AN"3<3:B=N&03B=[6¦]_;N&M=<=C5[\>UN&287C}@>9>"<4!2 m (t) = m = t1 ti ξ ξ ij ξ tn ξ ξ t t t t1 ,...,tn j k 1 n j,k=1 n t ξ ξ t t+τ 2 t s ξ 2 t+τ s+τ ξ ξ Bξ (t, s) = Bξ (t − s) . Fb N"_8:>"2;6°C87AN"3<3:B=N&03B=[\DtM303:32;C;CB32;M3032;03[\>"B3[6¦]2~CZ=< ,pQkZ3@tB=28M3032;03[\>"B=:"8: C87AN"3<3:B=N&03B=:"8:ªM303:32;C;C8NT:"0328Z3Z3@33<3:B3B=N&@?1vB3T33<3@ R(u) @3>"Z3lim @3287C}@?B32;R(u) M3032;03[\=>"B3R(0) :D?1vB3T=3<328D?:"7 u , u→∞ |R(u + ∆) − R(u)| = |M (ξu+∆ ξ0 ) − M (ξu ξ0 )| = |M (ξ0 [ξu+∆ − ξu ]| 6 q p p 6 M ξ02 M [ξu+∆ − ξu ]2 = R(0) 2(R(0) − R(∆)) → 0 ∆→0 M303< 7dN&T?TN&T 2 M [ξu+∆ − ξu ]2 = M ξu+∆ + M ξu2 − 2M (ξu+∆ ξu ) = 2(R(0) − R(∆)) . 1_1B32;0=N&>28B=C87&>N¬1:"ªC5T:8:SC5Z=24!v287A]w37&: |R(u)| = |M ξu ξ0 | 6 p M ξu2 q M ξ02 = p R(0) p R(0) = R(0) . M N&T3<=6 :"0=N&_;:6¦]UT:"030328Z3@33<3:B3B=N&@1vB3T33<@{C57dN&=<3:"B=N"03B3:8:SM=03:"32~C8C;Nk@3>"Z3@3287"C}@B328:7&03<3=N"7&25Z!`"B3::"M30=24!2A@ Z=2;B3B3:Dw32;7&B3:D{1vB3T=3<328D{"Z=287&>:"0U@3287B32;0=N&>28B=C87&>"v |R(u)| 6 R(0) , !prµ&3 '3&r#mU"^ tr#p3 'j' .1vUC57&`\B!N T = [a, b] _~NA4N"BkCZ=vw=N"D3B3[\DFM303:32;C;C ξ Nd@F1vB3T33<3@ f (t) ,~N"_8:"`"2~6 [a, b] 7&:"w3TN6< <{0=NC8C;6:"7&03"28Z=<3w3<3B=2~]=7&:7=87&:7 M303254!25Z%B=N"_8max [\>UN&287C}@<3∆tB37&2;80=N&Z=:6 :"79CZ!vUw!N&D3B3:8:9M303:32;C;C8N ξ <{:":"_;B=NAw=ζN"287"C}@C8<=6>":"Z=:6 16i6n i t t ζ= Zb f (t)ξt dt a b12~C5:"UC57&>28B3B=[\D<3B37"2880!NdZ x M30=< "2;B3B3[¦|<3B37"2880!NdZ=:>M=03< a → −∞a< =b −∞ , su|U:~4!<=6:C87&`<=B37&28;0=NdZ!`"B3[¦| C8v66 ζ →M3:"∞B3OC5Z=24vI>cS2~6C86[C5Z=2QJYC8vUª2;C87&>"v287RC5Z=vw=N&D=B=Nd@ >"28Z=<3w3<=B=N J 7dN&TNd@^]w37":qM303< n → ∞ x P=, M (ζ − ζ) → 0 . $, C}|# :~%-4!,<=7"C}# @/ T{KM37&032;25:"4!0325<3Z=<v 62;>q0=N{C86U:&N&Z=2;`7"TC}:ª@^]7&:w3}7&4:µN3M=]:TC:Z!}424!N:M3>03N&<7"25Z=`B3:C87&`?C5Z=vw=N&D=B3[¦|R>25Z=<3w=<3B ζ ζ n, m → ∞ xzP=,-+;gU M (ζ − ζ ) → 0 . =1'±p-³ ) #'&.,.' #) !-, , n 2 n n n ζ= Zb Zb a ! ' f (t)ξt dt R(t, s)f (t)f (s)dtds . a A=M *! , ,' # ( ' F62;2;6 Zb a +' ,' *% # % ' )&#' .,'&# ) ! , A= 2 m Zb f (t)ξt dt a 2 . / M (ζn − ζm )2 = M − 2M n X m X n X n X i=1 j=1 n X i=1 i2 f (ti )ξti ∆ti − m i2 X f (sj )ξsj ∆sj = f (ti )f (sj )ξti ξsj ∆ti ∆sj + M f (ti )f (τk )R(ti , τk )∆ti ∆τk − 2 i=1 k=1 m X m X + j=1 k=1 j=1 n X m X i=1 j=1 f (sj )f (σk )R(sj , σk )∆sj ∆σk . f (ti )f (sj )R(ti , sj )∆ti ∆sj + ~+ g" !42;C5` ti = τ i < sj = σ j ,s®: Z!N"C8B3:M3:C5Z=24!B=2;6v_~N"628w!N&B3 A= = = M N&T3<=6 :"0=N&_;:6¦]M303< lim max(∆ti ,∆τj )→0 lim max(∆ti ,∆sj )→0 lim n X n X f (ti )f (τj )R(ti , τj )∆ti ∆τj = i=1 j=1 n X m X f (ti )f (sj )R(ti , sj )∆ti ∆sj = i=1 j=1 m X m X max(∆sj ,∆σk )→0 f (sj )f (σk )R(sj , σk )∆sj ∆σk . j=1 k=1 UNS>7&n):"0=→:"8:∞v7&>"2;0U©ª4M28B3(ζ<@?7&−28:ζ032;)6[ →_;N06287&Nd@w=N"C87&`q0=N&>28B=C87&>NC57"032;6<37C}@?T A , QkZ=@{C8vUª2;C87&>":>N&B=<@B32~C5:C57">"28B=B3:";:<=B37&28;0=NdZNµx M30=< B3<3@B32~C5:"UC57&>28B3B=:"8:<3B37"2880!NdZ!N max ∆ti → 0 A= Z∞ Z∞ a = −∞ < b=∞ ®4!:UC57dN&7&:w3B3:C8vUª2;C87&>":>N @ R(t, s)f (t)f (s)dtds . −∞ −∞ ;0=NdZ K w=NC57&B=:C57"<^]4=Z3@C87AN"3<3:B=N&03B=:"8:kC5Z=vw=N&D=B3:";:kM303:32;C;C8NSC T:"030328Z3@33<3:B3B3:D1vB3T33<32;D ηt = M30=<ZK> ":U6 Z∞ <3B=7&2R@ h(t − s)ξs ds C5vS2~C57&>v"2;7~]2~CZ=<{T:"B=28w32;B<3B=7&28;0=NdZ −∞ t∈T Rξ (u) Z∞ Z∞ R(s1 − s2 )h(t − s1 )h(t − s2 )ds1 ds2 = Z∞ Z∞ R(s1 − s2 )h(s1 )h(s2 )ds1 ds2 . QkZ3@?CZ=vw=N"D3B3:";:ªM=03:"32~C8C;N ξ B=N>&|:~42Z=<3B=28D3B3:8:91<Z=`d7&0!Nª1vB3T3=<@?C80324!B=<|"[¦|:~42G1<Z!`d7&0=N −∞ −∞ −∞ −∞ t t ηt = <=62"> 79>"<4 Z∞ −∞ mη (t) = M h Z∞ −∞ i h(t − s)ξs ds = h(t − s)ξs ds Z∞ −∞ h(t − s)M ξs ds = Z∞ −∞ h(t − s)mξ (s)ds xzP=,-+"+A +;g < Rη (t, s) = M Z∞ Z∞ h Z∞ −∞ h(t − u)ξu du Z∞ −∞ h(t − u)h(s − v)M [ξu ξv ]dudv = Z∞ Z∞ i h(s − v)ξv dv = x P=,-+~I h(t − u)h(s − v)R(u, v)dudv . KL:"_~6:A©9B=:C57"`SM3:U628B@37&`q62~C57dN"6<{_8B!N&T{6N&7&2~6N&7&<3w=2;C5T:8::A©9<4N&B3<3@<<=B37&28;03<303:>N"B3<@B=2G>[\_8[\>UN&287C5:Q@ 6B32;B3<@^]A2;C5Z="9: =5Z=2~628B37dN&0=B3[¦|SC5:"[\7&<3DªC5:C87&:"<=7<3_LTU:B328w=B3:";:w3<=C5Z!N =8Z=2;62;B37dN&03B3[¦|kC8:""[\7"<3D ]3T:"7&:03[\21<=6A2 >c79>"280=:A@37&B3:C87&< p , i = 1, . . . , n , M :"}4)N Ω = {ω , . . . , ω } −∞ −∞ 1 n −∞ −∞ i mη (t) = n X ηt (ωi )pi = i=1 Z∞ = n X pi i=1 h(t − u) n X Z∞ −∞ h(t − u)ξu (ωi )du = Z∞ pi ξu (ωi )du = h(t − u)mξ (u)du . ¬1:030325Z=@33<3:"B=B=Nd@{1vUB=T33<@?M303:"=2;C8C;N96:A©ª287"[\7"`M=:&Z=vw32;B=N9N&B=N&Z=:";<3w3B3[6t:""0!N&_8:U6-J −∞ i=1 Rη (t, s) = M [ηt ηs ] = M = n X pi i=1 = Z∞ −∞ ∞ n XZ h Z∞ −∞ Z∞ Z∞ Z∞ X n i=1 −∞ −∞ h(t − u)ξu du Z∞ −∞ h(t − u)ξu (ωi )du i=1 −∞ −∞ = −∞ Z∞ −∞ i h(s − v)ξv dv = h(s − v)ξv (ωi )dv = h(t − u)h(s − v)ξu (ωi )ξv (ωi )dudv = pi ξu (ωi )ξv (ωi ) h(t − u)h(s − v)dudv = = Z∞ Z∞ M [ξu ξv ]h(t − u)h(s − v)dudv . N"C;C86:7&03<=6¡C5Z=vw=N&D^]!T:"}4N ξ @3>"Z3@3287"C}@µC87AN"3<3:B=N&03B=[6 M=03:"32~C8C8:6¦,=K+=87":6CZ=vw=N"2 m (t) = m ¦< ,=.1:UCZ=2GM3:~4C87dN&B3:>"T3< =87&<|?vCZ!:">"<=D>%x P!, ++~®N&@ w=NC57&`/> =87&:U60!N&>"2;B=C57">"2%_;N">"<=C8<37X7&:&Z!`"T:/:"7 t − s ]®M303<|U:~4!<=6T¡>"[\>":~4v"]Yw37":2;C5Z=< @3>&Z3@32;7"C}@q1vUB=T33<32;Dq0=N&_;B3:C87&< ]&7":k©ª2cC8N"6:2\>"2;03B3:F<:7&B3:C8<37&28Z=`"B3: , M :ª2;C57"`kM303:32;C;C R (t, s) B=NG>[¦|:~4!2LZ=<3B=28D3B3:8:F1<Z=`d7"0=Nc@3>&Z3@32;7"tC}@9−C8s7dN&3<3:B=N&0=B3[6¦]&2;C5Z=<9B=N1>&|U:~491<Z!`d7&0=NGRM3:~(t,4N"28s)7"C}@9C57dN&3<=:"B=N"03B3[\D M30=:"32~C8C&, !#1'¨±pz´ , "+ # ") ( ' ! , *+ ,.#'& )+,.) . ! .! #,#'& , #) ξ η "+ # )+#' ' ,"! # )+,.) . ! .! # / *! , ,' # ( '& N"62;B3<=6<=B37&28;0=NdZ%T:"B=28w3B3:D?C8v66:"D / ηtn = n X ξsi h(t − si )∆si . 1. :/C5>:"D=C87&>"v B3:"0!6NdZ=`B3[¦| C5Z=vw=N&D3B=[¦| >"28Z=<3w3<=B Z!<3B328D=B=Nd@¡T:6<3B=N&=<@ 7dN"T3<| >25Z=<=w3<3B@3>&Z=@3287"C}@B3:"0F@ 6N&Z=`"B3:0=N"C8M3032425Z=2;B3B3:"D,.1:N=87&:U6v @3>&Z=@3287"C}@µ5NAvUC8C5:>C8T3<=6 M303:"=2;C8C8:6¦,K®[\28032~6M3:C5Z=24:">N"7&25Z!`"B3:UC57&` 0=N"_8<328B3<=D {s } ] n = 1, 2, . . . ]37dN&ηT]w37&:"[ M (η − η ) → 0 , M :"}4N i=1 n t n k k=1 n t t 2 Rηn (u) = M ηun η0n → M ηu η0 = Rη (u) < mηn (t) = M ηtn → M ηt = mη = mη (t) E 7"CA>L4NC5Z=24!v287A]~w37&:¦4=Z=@kTN&©4!:";:\B=N&:"0=N\6:U628B37&:> B=N"&ZC>L428B3<3D B3:0=6NdZ!`"B3[\2YT:B328w3B=:6280=B3[\2 0=NC5M30=24!28Z=28B3<3@ F C5Z!N&:C}|U:~4=@37"C}@T{B3:0=6NdZ!`"B3:U6vqTU:B328w=B3:6t 2;,03. B3. .:,6tv0=N"C8M3032425Z=2;B3 F , K TNAw=2;C57">"21M303<=62;0=N90=NC8C86:7&03<=64!>v62;03B3[\D?>"2;T37&:"0?B=N"&ZC>L4!2;B3<3D%6:62;B37&:> (t , t ) ,=.1vC87&` n t1 ,...,tm 1 m t1 ,...,tm 1 P {ηtn1 !42;C5` → 2π 6 1 p u1 , ηtn2 |B| 1 p 6 u2 } = 2π |B n | Zu1 Zu2 −∞ −∞ Zu1 Zu2 −∞ −∞ 1 exp − (u − mn )(B n )−1 (u − mn )0 du1 du2 → 2 1 exp − (u − m)B −1 (u − m)0 du1 du2 = P {ηt1 6 u1 , ηt2 6 u2 } . 2 mn = (mnη (t1 ), mnη (t2 )), < 2 m = (mη (t1 ), mη (t2 )), Bηn (t1 , t1 ) Bηn (t1 , t2 ) , Bηn (t2 , t1 ) Bηn (t2 , t2 ) Bη (t1 , t1 ) Bη (t1 , t2 ) B= Bη (t2 , t1 ) Bη (t2 , t2 ) Bn = =, .1032425Z=`B3[\DM=280328|:~4%M3:~4{_;B=N&T:U6<3B37"2880!NdZ!N>":_;6:A©28B^, $p jUprE±³ suZ=vw=N"D3B3[\DM303:"=2;C8C B!N&_8[\>UN&2;7"C}@C57dN&=<3:"B=N"03B3[6/>S<=03:"T:6tC86[CZ!2"]2;C5Z=< 1vB3T3=<@?C80324!B=<| m (t) M3:UC57&:A@3B=B=N3]3NT:"030=25Z3@33<=ξ:"B3B=N&@?1vB3T3=<@ R (t, s) _;N&><=C5<=77":&Z=`TU::"7 |t − s| , E M=0324!28Z=28B=B3[\D0=N"B3282C57dN&=<3:"B=N"03B3[\DRM303:"=2;C8CSB=N&_;[\>QN > 7C87AN"3<3:B=N&03B=[6¡>v_8T:U6 C86[C5Z=2", 1> :"D M30=:"32~C8C&]~C57dN&=<3:"B=N"03B3[\DF>v_;TU:U6{C86[CZ!2Y@3>&Z=@3287"C}@kC57dN"3<3:"B!N&03B3[6?>ª<303:TU:U6{C86[CZ!2", E "0=N"7&B3:"2B=28>"2;03B3:!, NAvUC8C8:">C8T3<3DªM303:"=2;C8C\C87dN&3<3:B=N&0=B3[\Dª>Fª<303:TU:U6C;6[CZ=2\@3>"Z3@3287C}@C57dN"3<3:"B!N&03B3[6R1v_8T:6OC86[RC @ Z=2, ! # ! / .1vUC57&` u = (u1 , u2 ) t ξ ξ ξt = ξ cos λt + η sin λt , "+ ++ }4!2 < ¯B328T:030325Z!<303:">UN&B3B=[\2kCZ=vw=N"D3B3[\2F>"28Z=<3w3<=B3[F]4=Z=@µTU:7&:"0=[¦| M ξ = M η = 0 ] Dξ = Dη = 1 ]!N λ ¯M3:ξC87&:A@3ηB3B=N&@>25Z=<3w=<3B=N3, M N&T{TN&T R(u) = M ξt+u ξt == M ξ cos λ(t + u) + η sin λ(t + u) ξ cos λt + η sin λt = = M ξ 2 cos λt cos λ(t + u) + ξη(sin λ(t + u) cos λt + cos λ(t + u) sin λt) + η 2 sin λt sin λ(t + u) = = cos λt cos λ(t + u) + sin λt sin λ(t + u) = cos λu , 7&:9M303:32;C;C ξ C87AN"3<3:B=N&032;B>9ª<303:"T:U6C86[C5Z=2",=¬1:030328Z3@33<3:B3B=Nd@?1vUB=T33<@3=87&:8:9M303:32;C;C8Nª<=6282;7M=0324C@ C57dN">&Z=2;B3<32 t Z∞ Rξ (u) = }4!2 −∞ M303< x 6 −λ, M303< − λ < x 6 λ, M303< x > λ . !"(!n&(/$## =' !()'pr# l ) K¡T:B37&28TC87&2F7&28:03<3<{C5>"@3_8UN&B3<=D JM=0328:"0=N"_8:">UN&B3<=2 SvN@ 03`2 cos ux dF (x) ,   0 F (x) = 0, 5   1 X(f ) = <{:"0=N&7&B=:"2GM30328:"0=N"_8:>N&B=<32 Sv03`"2 Z∞ x(t)exp{−2πif t}dt −∞ x(t) = Z∞ X(f )exp{2πif t}df . 1. 280=2;628B=B3v > v842;6{7&0=N"T37&:>N&7"` TN&Tk>032;6#@^]ANM=28032~628B3B3v > TN"TFw=N"C87&:"7"v",;QªN&B3B!Nd@FM=N&0!NM3032;:"0=N&_;:">N"B3<3D <=C8M3:&Z=`_8v287"C}@tM=03<{:"M3<=C;N&B3<=<w=NC57&:7&B3:Q@ >032;62;B3B3[¦|C8:":"7"B3:"ªf28B3<3D?B32;M32803<=:~4!<3w32~C5T3<|?C5<3;B=NdZ=:>=, E M32;0=N&3 M3032;:"0=N&_;:">N"B3<@ Sv03`"2%v;4!2;6:":"_8B!NAw=N&7&` F{x(t)} = X(f ) ]YNX:"0=N&7&B=:"2¯tw328032;_ F {X(f )} = x(t) , N"M3<=C5` x(t) ↔ X(f ) :"_;B=NAw=N"287A]=w37": X(f ) M3:&Z!vUw!N&287C}@{M=03<=628B=28B3<32~6 M30=28:"0=N&_;:">UN&B3<@ SvU0=`"2STR1vUB=T33<3< ]\N ¯ 0328_;v;Z!`d7AN"7M303N"B3<@ kv03`2µT ) ,uQkZ3@¡B=N"C x(t) ¯ >"2;S2;C87&>"2;B3B=N&@ x(t) 1vB3T33<3x(t) @^]"N X(f ) ¯kT:6MZ=2;TC5B3:_8B=NAw3B!Nd@ª1vB3T3=<@^,"KM=:&Z3@303B=:"D91:"0=X(f 62C8M328T37"0 X(f ) 6:A©qB3:1_;NA4)N&7&`1w32;0328_ 28;:qN"6MZ=<=7&v;4!B3v ><%FN&_;:">"v > |UN&0!N&T37&2;03<=C87&<3T3< −∞ −1 OuC5Z=< X(f ) = |X(f )|eiθ(f ) . x(t) M=03<3B3<=6N"2879>"2;S2~C57&>28B3B3[\21_;B=NAw328B=<@^]7&: X(−f ) = X(f ) = |X(f )|e−iθ(f ) . .1:Q=;7&:6v%N6MZ=<37&v;4!B=[\DXC8M328T=7&0µ¯ =87":w32;7&B=N&@R1vB3T33<@RM328032~628B3B=:"D f ])NFN&_;:">N&@µ1vUB=T33<@%¯%B32;w3287&B!Nd@ 1vB3T33<3@:"7 f , OuC5Z=< x(t) ↔ X(f ) ]7&: F{x(t − t0 )} = Z∞ −∞ x(t − t0 )e −2πif t dt = Z∞ −∞ x(µ)e−2πif (µ+t0 ) dµ = X(f )e−2πif t0 . < ++~I F{x(t)e 2πif0 t Z∞ = x(t)e 2πif0 t −2πif t e dt = Z∞ x(t)e−2πi(f −f0 ) tdt = X(f − f0 ) . M N&TkB!N&_8[\>UN&2~6Nd@ @ 1vB3T3=<@F<Z=N1vB3TF@ 3<=D ϕ (x) = ϕ ] δx ∈ R ]M303< ε → 0 ])}4!2 ϕ ¯{ZC>cNd@OB328:7&03<3!N&7&28Z=`"B=N&@<3B37"2880=<303v"2~6Nd@X1vUB=T33<@^] v;4!:>&Z=2;7&>":0@Fc> ªN&@C5:":7&B3:S28B= R ϕ(t)dt = 1 , F6M=v;Z!`C5B!Nd@?1vB3T=3<@:&Z!Nd4N&2;7ªC8>":"D!C57&>UN"6< −∞ x ε 1 ε ε −∞ ∞ −∞ Z∞ −∞ Z∞ x(t)δ(t − t0 )dt = lim ε→0 −∞ x(t)ϕε (t − t0 )dt = x(t0 ) , F{δ(t)} = F −1 {δ(t)} = 1 . 1B=7&28;0=NdZ%C5>28037&T=<4>"vU|1vB3T33<=D < h(t) x(t) Z∞ y(t) = <Z!qC8:"T30=N"S2;B3B3:"D_;N"M3<=C5< OuC5Z=< x (t) ↔ X (f ) < x (t) ↔ X (f ) ]37": 1 1 2 x1 (t) ∗ x2 (t) = −∞ h(t − u)x(u)du y(t) = h(t) ∗ x(t) . 2 Z∞ x1 (τ )x2 (t − τ )dτ ↔ = Z∞ −∞ :"M30=24!28Z3@3287"C}@?TN&T x1 (τ ) Z∞ Z∞ Z∞ −∞ −∞ x2 (t − τ )e x1 (τ )x2 (t − τ )e−2πif t dτ dt = −2πif t dtdτ = X2 (f ) Z∞ x1 (τ )e−2πif τ dτ = X1 (f )X2 (f ) . us Z=254!:">UN&7&28Z=`"B=:=]:"M32;0=N&3yC8>"280=7&T3":?>"032~628B3B3:DR:&Z!NC57&{w=N"C87&:"7&B=:"D :&Z!NC57&< x (t) ∗ x (t) ↔ X (f )X (f ) . KL>"<4!vFC5<=662;7&03<3<9M=N"03[M3032;:"0=N&_;:">N"B3<3D Sv03`"2]~v6B3:A©28B3<=2u>":G>"0=2;628B=B3:"Dk:&Z!NC57&<ªM3280=25|:~4<37>1C5>"2;037&T3v >9w=NC57&:7&B3:"D?:""Z!N"C87&< (t) ↔ X (f ) ∗ X (f ) . M 28:032;6N.FN&0=C828>UNdZ3@9v7&>"2;0©4N"28x7A](t)x w37&: =8B32;038<@?C5<3;B=NdZ!NS>":k>032;62;B3B3:D:""Z!N"C87&<0=N">"B=1N =;B3280=8<39w=N"C87&:"7&B=:"D?:"&ZN"C57"< −∞ −∞ 1 2 1 Ex = −∞ 1 2 Z∞ 1 x2 (t)dt = 2 2 Z∞ |X(f )|2 df . 3 '"l"=()&#rpn¨$p!²#" $p Y3j Upr −∞ −∞ KL2;03B32;6C}@RTR0=NC8C;6:"7&032;B3 C57dN"3<3:"B!N&03B3:8:5NAvUC8C8:">C8T:"8:qC5Z=vw=N&D3B=:"8:M303:32;C;C8N3,KL>"254!2;6 M30328:"0=N?@ _;:">N"B3<32 Sv03`2 F{R (τ )} = C (f ) < F{R (τ )} = C (f ) T:"030328Z3@33<3:B3B3[¦|¡1vB3T33<3D R (τ ) < R (τ ) M30=:"32~C8C5:> ξ < η B!Nª>"|:~4!2G<{>"[¦|U:~4!2 Z!<3B328D=B3:";:91<3Z=`d7&0=N ξ t ξ η η t η(t) = CG<=6M3v~Z=`UC5B3[6 :7&TZ=<3T:U6 , h(t) Z∞ −∞ h(t − u)ξu du ξ η w32;0328_ OuC5Z=<µ>C8:":7&B3:"ª28B3<=2 >U62;C87&: (4.13) 3 ] & 7 q : = " N " > ; 2 = B 5 C & 7 > ª : = M 3 = < 6 8 2 q 7 ><4 C (f ) ξ Rη (τ ) = Z∞ Z∞ Z∞ −∞ −∞ −∞ Z∞ = M3:~4C87dN&>"<=7&` τ <%>q<3B37&2;80=N&Z=2SM=0324C87dN&>"<=7&` t−s +"+~H Rξ (τ − µ − ν) Cξ (f ) exp(2πif (τ + µ − ν))h(µ)h(ν)df dµdν = Cξ (f ) exp(2πif τ ) Z∞ h(µ) exp(2πif µ)dµ Z∞ h(ν) exp(−2πif ν)dν . 1. 032;:""0!N&_8:>N"B3<32 Sv03`"2 F{h(τ )} = H(f ) B=N&_;[\>N"287"C}@M32;0324)N&7&:w3B3:"D 1vUB=T33<32;DZ=<3B=28D3B3:8:X1<Z=`d7&0!NC <=6M3v~Z=`UC5B3[6 :"7"TZ=<3T:6 h(τ , M N&T{TN&T F {h(τ )} = H(f ) ]=7&: ) −∞ −∞ −∞ −1 Rη (τ ) = E 7"CA>L4NM3:&Z=vw=N"2;6 Z∞ x P!, +;PU Cξ (f )|H(f )|2 exp(2πif τ )df −∞ x P=,-+~V M N"T3<=6y:"0=N&_;:6¦]>":_4!2;D=C57">"<32?Z=<3B=28D3B3:8:/1<Z=`d7"0=NOB=NO5NAvC8C8:">UC5T3<3D C57dN&3<=:"B=N"03B3[\D M303:"=2;C8C{M=:&Z=B3:UC57&`N> :"M=<=C5[\>UN&287C}@C5::"7&B3:S2;B3<32;6 (4.15) , QkZ3@%C57dN"3<3:"B!N&03B3:8:ªM303:"=2;C8C;N Cη (f ) = Cξ (f )|H(f )|2 M ηt2 = Rη (t, t) = Rη (t − t) = Rη (0) , M3:N=87&:U6v R (0) <3B37"2803M30=287&<30=v"287C}@OTN&TX6N&7"2;6N&7"<3w32;C8T:"29:A©q<4N&B=<326:SB3:UC57&28B=C87&>%x P!, +;PU® B=N&_;[\>N?>c7cC8M328T37"0=NdZ=`B3:"DªMZ=:7&B3:C87&`Q>¡6:SB3:UC57&<ªM303:"=2;C8C;N <Z=< =8B=2803;287&<3w=2;C5T=<=6 C5M=28T37&0=:6¦,=.1:A@=C8B3C<=6t(fC8)TN&_;N"B3B3:2",N"C8C;6:"7&0=<=6<42;NdZ!`"B3[\D?M3:&Z!:C5:>":D1<Z=`d7"0?CGw=NC5ξ7&:7&B3:"D|UN"0=N&T=7&2803UN&7&`qTN&T%vUC503254!B32;B3B3v >6:SB3:UC57&`M=03:"32~C8C;N >v_8T:D?M=:&Z=:UC52kw=NC57&:7 S<=03<3B3:D R∆f(0)C¦32;B37&03:U6X> f = (f + f )/2 ]dM303<=&Z=<3_;<37&28Z=`"B3:S0=N&>B3v > 2C (f )∆fξ,U.1: =;7&:"DqM=03<3w3<3B=2 C (f ) <3B37"2803M30=287&<30=vI>c7TN&T?0=N"C8M303254!25Z=2;B3<321M3:qw=NC57&:7&2FC503254!B32;D{6:SB3:UC57&&Z3@32;7"C}@w32;7&B3:D{1vB3T=3<328D, #j Uprq±pz #! ' ! , ( , '&'&(" ' ! # , −∞ ) ! / Cξ (f ) # ' ! ,.'&# ( , ++8P *! , ,' # ( '& OuCZ=<"[ C5>:"D=C87&>":F[¦Z=:SB=28>"2;03B3[6¦]U7&:ª6:A©qB3:S"[¦Z=:k[¡>"[\0=N&7&`Fw!N"C57":"7&[ <42;NdZ!`"B3:8:M3:&Z=:UC5/ :>":";:S1<Z=`d7"0=N9C1w=N"C87&:"7&B=:"D|UN&0!N&T37&2;03<=C87&<3T:"D?P=,-+;ª7dN&T^]3w=7&:"[ Zf2 f1 < f2 4=Z3@ Cξ (f )df < 0 . f1 £®7&:9M30=:"7&<3>:"032;w3<379B32;:"7&03<==N&7&28Z=`"B=:C57"< M ηt2 Zf2 ]7AN"T{TN"T Cξ (f )df = 1 1 Rη (0) = M ηt2 . 2 2 f1 .1vUC57&`q7"28M32;03`q5NAvC;C5:">UC5T3<=DM303:32;C;C M3:~4N"287"C}@?B=Nq>&|U:~44>"vU|Z=<=B328D3B=[¦|{1<Z!`d7&03:">!]3C5:24!<3B=28B3B3[¦| M=N"0=NdZ=Z=25Z=`B3:=,N"C;C86:7&03<=6 >"2;T37&:"0?>"[\:"03:ξw3B3[¦|_;B=NAw328B=<3D t W (u, v) = (ηu1 , . . . , ηuk , ζv1 , . . . , ζv` ) , M3:"Z=vw328B=B3[\DµB=N"&ZC>L4!2;B3<32~6 M=03:"32~C8C;N η B=N>[¦|:~4!2FM=2803>:"8:1<Z=`d7&0=N>6:62;B37&[ {u } "[¦|:~42G>7&:"0=:"8:1<3Z=`d7&0=Nq>6:U628B37"[ {v } ,K®2;T37&:0 W M3:&Z=vw32;B%>032;_8v~Z=`d7dN&7&2kZ=<3B32;D3B3[¦|{M30328:"0=N?@ _;:">N"B3<3DRM303:"=2;C8C;N ξ ,^.1Q: =87":6v W @3>&Z3@32;7"C}@R5NAvUC8C5:>C8T3<=6>"2;T37&:03:6¡4=Z3@ZK> "[¦|O6:628B=7&:">{B=N"&ZC>L4!2;B3<@ < , {u } {v } £®7": v7&>"2;0U©ª428B3<32u:C57dN"287"C}@F>"2;03B3[6?<14=Z=@k>"[\":03T3< W <3_®M303:"=2;C8C8:">\B=N >"[¦|U:~4!2 N > 2 M=N"0=NdZ3Z!25Z=`B3: C8:"24<3B328B=B3[¦|1<Z=`d7"03:">!, u k i i=1 ` j j=1 v t i j $p jUpr !´ s®:">":T3vM3B3:UC57&`{<3_ CZ!vUw!N&D3B3[¦|OM303:32;C;C5:">{B=N"_8[\>UN&287C}@RC5:>62;C87&B3:%5NAvUC8C5:>?@ 8C TU:D^]2;C5Z=<ZC> :"D?>"28T=7&:"0^]=M3:C87&03:"2;B3B3[\D?M3:~4!:"NB3:>"2;T37&:"0=v W ]3:"TN"_8[\>N"287"C}@C8:">U62;C57"B3:95NAvC8C8:">UC5T3<=6¦, ¬FN&©4![\Dq<3_\4>"vU|ªM30=:"32~C8C5:> < :"M=0324!28Z=28B]"2;C5Z=<<3_;>"2~C57&B3[ 1vB3T=3<3<C50=24!B3<3|9<TU:030328Z3@33<3:BL@ B3[\2k1vB3T=3<3<&=87&<3|?M30=:"32~C8C5:>=, E 4!B!ηN&T:ª4=ζ Z3@%7&:";:=]w37":""[ _;Nd4N&7"`qMZ=:"7"B3:C87&`C5:>62;C87&B3:8:90=N"C8M303254!25Z=2;B3<@ >2803:A@37&B=:C57"28DF>"2;T37&:0=N W B32;:"A|:~4<=6:\_8B=N"7&` T:>N&0=<=N&3C< >/ZC> :"DFM=N"03[/>"2;T37&:03:"> η , ζ , M N&T3<=6µ:""0!N&_8:U6¦] w37":""[¨M303:32;C;C5[ η < ζ [¦Z=<{:"M303254!25Z!28B3[ C5:">U62;C87&B3:ª4!:&Z=©9B=[¨[\7&`9_;NA4)N&B3[ T:>N&0=<=N&3<= ":D9M=N&03[6:62;B37&:>kB=N&"ZC>L4!2;B3<3D (ui , vj ) ,£®7&<94!:M3:&Z=B=<37&25Z!`"B3[\2C8>"2428B3<@qC5:~4280U©N"7"C}@ª>S1vB3T=3<3< Rηζ (u, v) = M (ηu ζv ), T:"7":"0=N&@B=N&_;[\>N"287"C}@C8:">U62;C57"B3:9TU:030328Z3@33<3:B3B3:"D{1vB3T33<=28D?M303:"=2;C8C8:"> Rηζ (u, v) = M Z∞ −∞ Z∞ Z∞ −∞ −∞ ξu−α hη (α)dα Z∞ −∞ ηu < ζv ξv−β hζ (β)dβ = M (ξu−α ξv−β )hη (α)hζ (β)dαdβ = Z∞ Z∞ −∞ −∞ Rξ (u − α, v − β)hη (α)hζ (β)dαdβ +"+AV QkZ3@{C57dN&3<=:"B=N"03B3:";:ªM=03:"32~C8C;N ξ <=62;2;6 t Rηζ (u, v) = Z∞ Z∞ −∞ −∞ Rξ (u − v + α − β)hη (α)hζ (β)dαdβ = = Z∞ Z∞ Z∞ −∞ −∞ −∞ = M N"T3<=6 :"0=N&_;:6¦] R Cξ (f ) exp(2πif (u − v + α − β))hη (α)hζ (β)dαdβdf = Z∞ @3>"Z3@3287"C}@{1vB3T33<32;D{0=N&_;B3:C87&< (u, v) −∞ ηζ Cξ (f ) exp(2πif (u − v)) Rηζ (τ ) = Z∞ Z∞ hη (α)e 2πif α < u−v dα Z∞ −∞ −∞ hζ (β)e−2πif β dβ df . xzP=,-+Aa" Cξ (f )Hη (f )Hζ (f )e2πif τ df . 1. :Q=;7&:6vªT:"030328Z3@33<3:B3B3[\2 1vB3T=3<3< ] < _;N&><=C}@3717&:&Z!`"T:F:"7F0=N"_8B3:UC57&<N"038v62;B37&:">F2803:A@37"B3:C87&28D/>28T37&:0=RN WR B=2_;RN">"<=C8<37%:"7B=NAw=NdZN%:"7"C8w3287dN>"032~628B3<^]#2;C5Z=<M30=:"32~C8C ξ B!N >&|U:~4!2®1<3Z=`d7&03:>GC87AN"3<3:B=N&032;B^,&K/7dN&T:U6µCZ=vw=N"2¦M303:"=2;C8C8[ η < ζ B=N&_;[\>NQ> 7"C}@ªC8:">U62;C57"B3:GC87dN&3<3:B=N&0=B3[6< M303:32;C;C8N6<^, s®TN&_~N&B3B=:"2 :UC57dN&2;7"C}@>2803B3[6X4=Z3@?C8:">:"T3vM3B3:UC57&<<3_ N CZ!vUw!N&D3B3[¦|M303:"=2;C8C8:">ªB=NS>[¦|:~4!2 Z=<3B32;D3B3[¦| 1<Z=`d7"03:">!]C8:"254!<3B32;B3B3[¦|M=N"0=NdZ=Z=25Z=`B3:=, N"C;C86:7&03<=6 >N&©qB3[\DRw!N"C57"B3[\D/CZ=vw=N"D^]^T:"}4N?B3:UC5<37"25Z=c7"C}@^]7&:{2;C87&` H (f )H (f ) = 0 , M :"}4N?<3_%xzP=,-+Aa& M3:"Z=vw=N&2~6¦]w37&: R (τ ) = 0 4=HZ3@X(f>)C825| Hτ , (f1_) C57dN"3<3:"B!N&03B3:UC57&S4!:&Z=©ª28B?"[\7"` B3v~HZ=2;>"(0)H :"DF:7&(0) TZ=<==TSB=0N\M=:C57":A@3B3B3mv > (0)m C8<38B=N&(0)Z!N3, E 7"CR>L4)N CZ!24!v287A];w=7&:c1vB3T3=<@kT:">N"03<=N"3<3"B=N¦B=v;ZK>S, M N"T{TN"T ξ @3>"Z3@3287"C}@?5NAvC;C5:">UC5T3"2;T37&:0=N < ζ(v) = (ζ , . . . , ζ ) η(u) = (η , . . . , η ) B32;_;N&><=C562~C57&B=:"8:q0!N"C5M=0324!28Z=28B=<@ p = p 0=N">"B=N9M=03:"<3_;>"254!28B3C< > MZ=:7&B3:UC57&2;D xzP=,-+;U (β) . p (α, β)) = p (α)p OuC5Z=<4!Z3@>UC525|6:U628B37":">9B=N&"CZ >L4!2;B3<3D {u } < {v } >"[\M=:&Z=B@32;7"C}@vUCZ=:>"<329x P!, +~]7&:M303:32;C;C5[ η < B=N&_;[\>?N >c7"C}@?C87dN&7&<=C87&<3w32~C5T3<{B32;_;N">"<=C8<=6[6<^, ζ 0ªjon '"l"=()&#rpn Xl^ M 280=6<3B ªv6 >k7&2;:"03<=<C5>"@3_8<:":_8B=NAw=N"2871B328©ª25ZN&7&28Z=`"B3[\.2 =5Z=2;T37&03<=w32;C8T3<321C5<3;B=NdZ=[F]UT:"7&:03[\2 >UC5A2 @ }4NcM303<=C8v7"C57&>v > 71.> =8Z=28T=7&03<3w32~C5T3<3|C5c0=N"_5Z=<=w3B=NG<ª>TCZ >cw=N&2;7cTN&Tª2;C87&2;C87&>"2;B3B3[\2"] 7dN&T%<µ"28B=B3[\2k<=C87&:"w3B=<3T3<^, FC5T3vUC8C87&>"2;B3B3[\2FSv6[¨¯ =;7&:ªv6[°<=C8T3vC;C57&>28B3B=[¦|?<=C87&:"w=B3<3T:"7> =8Z=28LT @ 7&03:U6N&;B3<37&B3:8:?<3_5Z!vUw=28B3<@^, O®C57&2~C57&>28B3B3[\2qªv6[ <=C}|U:~4=@37:"7{N&76:C812;03[F]C5:"Z=B3=N<µ403v8<|5N&Z!N&T=7&<3w3A2 @ C5T=<|S<=C87&:"w3B=<3T:">=, NAvUC8C5:>":0!N"C5M=0324!28Z=28B=<32¦w=N"C87&:1<=C5M3:"Z=`"_;v"2;7"C}@STN&T6:~4!25Z=`GSv6NG>FC5<=C87&2;6N&|]dM3:C8TU:"Z=`"T=v C5: Z!N"C8B3:=28B37&0!NdZ=`B3:"DRM3032425Z=`B3:"DR7&28:032;629M303"2~C5`U6N:"S<|%vUCZ=:>"<@|%0=NC5M303254!25Z!28B3<329>2803:A@37&B=:C57"28D ":"Z=`"ª<|µC5v66 C5Z=vw=N&D=B3[¦|µ>"28Z=<3w3<3BR>M=0324!28Z=29M3:~4!w3<=B@3287"C}@R5NAvC8C8:">:6v0!N"C5M=0324!28Z=28B=F< >S,.1v7&2;6¨1<Z=Q` @ 7&0=N"3<3<< =8T30=N"B3<303:>N"B3<@X>&Z=<@3B=<32ªv6:>%B=NµM=:&Z=2;_8B3[\DC8<38B=N&Zt6:A©qB3:C5B=<3_8<37"`<Z="v287F:~4!<3Bq2~C57&2~C57&>28B3B=[\DSv6¦]UB=N&_;[\>N&2~6[\D97&28MZ!:">"[6¦]UT:"7&:03[\DqvC87&0=N"B3<37&`ªB325Z=`_5@^, M 28M3Z=:">:"D ªv6 >"[\_;[\>N"287"C}@?7&28MZ!:">"[6t4!><©ª2;B3<32~6 =8Z=28T=7&03:"B=:">9B> =5Z!28T37&0=<3w32;C8T3<|{M303<3:"0=N&|, M 2F©!2 =5Z!28T37&0=:"B3[F]!TU:Q@ 7&:03[\2:7&>"2;w=?N >c7F_;N =5Z!28T37&0=:"M303:>":~4!"Z3F@ > 7"C}@qM30=<3w3<3B3:D7&2;MZ=:">:";:Gªv6N3, M 28MZ!:">":DqSv6X6:A©qB3: :M3<=C8N"7&`1TN&TqC57dN&=<3:"B=N"03B3[\DqCZ!vUw!N&D3B3[\D9M30=:"32~C8C\C\B3v~Z=2;>":D91vB3T=3<328DqC80324!B=<|,KL:k6B3:";<|ªM=03<Z=:A©28B3<@| 7&2;:"03<=&@3_;N&7"`C}@RC9<=C87&:"w=B3<3TN"6%T:"7&:03[¦|RC5M=28T37&0!NdZ=`B=Nd@ MZ!:"7&B3:UC57&`R6:"SB=:C57"UC5T:";:Sv6N3]pB=N&T3Z!NA4![\>U?N >cS28;:C}@OB=N%M3:"Z=28_;B3[\D/C5<3;B=NdZ]p:C87AN"287"C}@/M30!N&T37&F< @ w32~C5T3"MZ!:"7&`µ4!:w=NC57&:7A]6B=:"8:R":"Z=282>[C5:T3<|]pw=2;6 w=NC57&:7&[F]#@3>&Z3F@ >cS<32~C}@/:C5B=:">"B=[6<> C;N"6:6¡C8<38B!NdZ=2,.1N: =87&:U6v?M=0324!M=:&Z!N"5N&287C}@^]=w37&3: =8B=2803;287&<3w=2;C5T=<3DRC8M328T37"0Sv6N@3>&Z3@32;7"C}@µM3:UC57&:A@3B3B=[6B=N >UC525|w!N"C57":"7dNd|<{>>":~4!<37C}@M3:"B3@37&<32F"28Z=:";:ª5NAvUC8C8:">NkSv6N n TN&T%C57dN"3<3:"B!N&03B3:8:ª5NAvC8C8:">UC5T:";:kM303:32;C;C8N CGB3v~Z=2;>"[6 C503254!B3<=6 <%C1M3:C87&:A@3B3B3:DC5M=28T37&0!NdZ=`B3:"D?MZ=:7&B3:C87&Q` >6:SB3:UC57&< t Cn (f ) = N0 /2, −∞ < f < ∞ . M N&T:"D?Sv6¦]3v84vUw=<M30=:"M3vS2;B3B3[6 w3280=28_1Z=<3B=28D3B3[\D%1<Z!`d7&0^]U4=Z3@{T:"7&:03:";: Z∞ |H(f )|2 df < ∞, M30=28>"0!N&ªN"287"C}@B!Nª>[¦|:~4!2G1<Z=`d7&0!NS>qC57dN"3<3:"B!N&03B3[\D5NAvC;C5:">UC5T=6DqM=03:"32~C8C B3<=2;6¦, 1_10=N">"28B!C57&>UNxzP=,-+~V"®M3:"Z=vw=N&2~-6 J −∞ Cµ (f ) = :7"CR>L4N9C5Z=24!v287A]w37&:6:SB3:UC57&`9M30=:"32~C8C8N0=N">"B=N M µ2t µt CGB3v~Z=2;>"[6tC80324B3<=6 _8B=NAw32A@ N0 |H(f )|2 , 2 N0 = Rµ (0) = 2 Z∞ |H(f )|2 df . ¬1:030325Z=@33<3:"B=B=Nd@1vB3T=3<@M=03:"32~C8C;NªB!Nª>[¦|:~4!211<Z=`d7"0=N<=6282;7><4 N0 Rµ (τ ) = 2 Z∞ −∞ x P!, +~ |H(f )|2 cos(2πf τ )df . 1¬ :030325Z=@33<3:"B=B=Nd@S1vB3T3=<@S"28Z=:";: ªv6NGM3:"Z=vw=N&2;7"C}@kB32;M3:C80324C87&>"2;B3B3:G<3_¦>"[\0=N&©ª2;B3<@14=Z3@9C5M32;T37&0=N&Z=`"B=:"D MZ!:"7&B3:UC57&<%6:SB3:UC57&L4NC5Z=24!v287A]Lw37&:ZK> "[\2R>"[\:"03:w3B3[\2%_8B!NAw328B3<3@ "25Z!:"8:/5NAvUC8C8:">NRSv6N@3>&Z3@F>c7"C}@ C57dN&7"<=C57&<=w32;C8T3< B32;_;N">"<=C8<=6[6<^]&TN&T9[ "Z=<3_8T:FB32 >"[\"<=0=NdZ=L4!28B3<3@^,K =;7&:6XC;6[CZ=225Z=[\D95NAvUC8C5:>?@ C8T3<3D/Sv6 :"M=<=C5[\>UN&287%M=0324!28Z=`"B=vI> CZ=vw=N"D3B3:C87&`=,p.1:~4C57dN&>"Z3@@ R (τ ) >tx P!, +~H1M=:&Z=vw=N"2;6¨M30324)C57dN&>"Z=28B3<=2 T:"0=0325Z3@3=<3:"B3B=:"D{1vUB=T33<3<{M303:32;C;C8NB=N9>[¦|:~4!2cZ=<3B32;D3B3:8:1<Z=`d7"0=N3]B=N9>"|:~4%T:"7&:03:";:ªM3:~4)N&287C}@"28Z=[\D ªv6 n N0 Rn (τ ) = 2 Z∞ Z∞ −∞ −∞ N0 δ(τ + u − v)h(u)h(v)dudv = 2 Z∞ −∞ h(u − τ )h(u)du . "+ +da N628B@@:M3280!N&3 C8>"280=7&T3<>:ª>032;62;B3B3:D:""Z!N"C87&<v6B3:A©28B3<=2;6>w=NC57&:7&B3:"D:""Z!N"C87&<^]M303<|U:~4!<=6C8B3:">UN T{0=N&>28B=C87&>"vµxzP=,-+;U, K TNAw=2;C57">"2GM303<=62;0=N90=N"C;C86:7&03<=6 <4!2~NdZ=`B3[\D?1<3Z=`d7&0?B3<©qB3<|?w=N"C87&:"7qC1M32;0324N"7&:"w=B3:"D{1vB3T33<32;D ( 3M 0="|:~4 =87&:8:q1<Z=`d7&0!N9M3:C87&vM=N&2;725Z=[\D{5NAvUC8C5:>C8T3<3Dªv6 M303:32;C;C8NB=N>"[¦|U:~4!2 µ :"M=0324!28Z3@3287C}@0=N">"28B!C57&>:6 W (f ) = 1, |f | < W, 0, nu ]37&:1vB3T=3<@%C503254!B3<| mµ (t) t mµ (t) = Z∞ mn (u)h(t − u)du . 1b :SM=:ª:M3032425Z=2;B3 M303:"=2;C8C;N 1vB3T33<3@C503254!B3<| 0!N&>"B!NFB3v~ZC>S]7dN"Tw37&: ,¬1:030328Z3@33N&0=<=N&3<=:"B3B=N&@1vB3T3n3<3"[¦|Um:~4!2G(u)_;NA4)N?> 7C}@?C5::"7&B=:"S2;B3<32~6 m (t) = 0 −∞ u n B(τ ) = R(τ ) = = N0 2 ZW Z∞ µ Cµ (f )|W (f )|2 cos 2πf τ df = −∞ cos 2πf τ df = N0 sin 2πf τ 2 2πτ suZ=254!:">UN&7&28Z=`"B3:!]6:"SB=:C57"`M=03:"32~C8C;NSB=N9>[¦|:~4!2 µ ]30=N">"B=N&@ −W W −W = W N0 t M3:UC57&:A@3B3B!N3, Bµ (0) = Rµ (0) = M µ2t = W N0 , sin 2πW τ . 2πW τ ++; i + ,!s®28>NC57&`"@3B3:"> SG, ¥S,=¬1v0=CG7&28:03<3<{>"2;03:A@37&B3:UC57&2;D<%6N&7"2;6N&7"<3w32;C8T:"D{C57dN&7&T:">ª¥S, ¥S, M 2;:"03<3@?>2803:A@37&B=:C57"28D^,3/, J£Y4!<=7&:"039KG, .9,3¬1v0=CG7&2;:"03<3<{>2803:A@37&B=:C57"28D^,3/, J=bFNAvTN3]+;Ua, P!", ! <30U@32;>9¥S, b9,!K®2;03:A@37&B3:UC57&`!,/, J!bFNAvUTN=]+~"3, V3, 1>"w32;B3T: , 9, ]3µ24!>24!2;$> # , 9,=µN&7"2;6N&7"<3w32;C8TNd@?C87dN&7&<=C87&<3TN3,!/, J!K®[C8ªN&@ST:&ZN3]+;I3, =F, S®:03:">T:">ª¥S, ¥S,=µN"7&2;6N"7&<3w32~C5TNd@{C57dN&7&NC57&`"@3B3:"> Sc, ¥k,-F] G1<=C87;@3T:>KG, .9, ]3)v"T:>ª¥k, /,!s®:"03B3<=TM3:7&2;:"03<3"2;03:A@37&B3:UC57&2;D^,3/, J3bFNAvTN=]!+~"g3, =, 1>w328B3T: , 9,-])µ24!>24!2;%> # , 9,-] G1<=C57~@3T:">¥S, KG,s®":03B3<3T_;NA4)NAwµM3:6N"7&2;6N"7&<3w32~C5T:"DOC57dN&7"<=C57&<=TU2,/, J KL[C5Nd@ªT:&Z!N=]+;"3, ' & =, B3254!28B3T:7SG, KG,3¬1v0=CG7&2;:"03<3<{>2803:A@37&B=:C57"28D^,3/, J=bFNAvTN3]+;"3, +~g3, 6v0=6N"B{KG, O, M 28:"0=<@?>2803:A@37"B3:C87&28D?<%6N&7&2~6N&7&<3w=2;C5TN&@{C87AN"7&<=C87&<3TN3,3/, J=KL[C5Nd@?ST:&Z!N=])+;"3, ++",5:"_~N&B3:># , ¥k, M 2;:"03<@k>2803:A@37"B3:C87&28D^]~CZ=vw=N"D3B3[\2YM=03:"32~C8C8[R"2;03:A@37&B3:UC57&2;D<{2821M30=<Z=:A©ª2;B3<@^,7A,+,3/, J=%<30^]+;""P=, +~H3,¬10=N"62;0 ,3µN&7&2~6N&7&<=w32;C8T3<32k6287":~4![ C87dN&7&<=C87&<3T3<,!/, J3%<30^]+;Ua"V, +;P=, 6vU0!6N&BFKG, O,~YvT:">:~4C57&>:®TF0=28S2;B3F< >_~NA4NAw1M3:\7"28:"0=<3"2;03:A@37&B3:UC57&2;DG?¥S, KG,-]pWcªN"TU:>µKG, ,-]WcªN&T:>µb9, ,NA4NAw=2803:A@37"B3:C87&28D J E C5B3:>"B3[\2qM3:B@37&<@^, .103254!25Z!`"B3[\2G7&28:032;6[F,)suZ=vw=N&D3B=[\2FM303:"=2;C8C8[F,3/, J=bFNAvTN=]!+~"3,