А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 17
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.]j\*Njm*IW.DJNj\*lOHIHnmD&OHO=DK`FHI]j\*Njm*IW.DJNj\*lOHI]^J.l)G=D&]rV.Itm*QJ]rKTI]PYI\*lYb JIkHYDW_-vG=D&O=DuFHGHIQstWI\*lOHIHnFHI]j\*Njm*IW.DJNj\*lOHI]^JlG=D&]rV.Itm*QJ]Kω0ξ1 (ω), ξ2 (ω), . . .W5YDM5m*Id'JIkHYNnHkJIzQysLD&WNPGpD&N^J9GHN^pNPOHQHNXsLDmDLkHQω∈Ω¾x%ÄSxax» £H=IYDsLDJlHnHkJI5NP]j\*QQnJIPPξn −→ ξ ξn −→ ηP(ξ = η) = 1ÁaCE -ÄSI]^JDJIkHOHI)FHIYDsLDJlHnkJImH\=Kq\*fcv_:VP(|ξ − η| > δ) < εQ*sXOHNPG=D&WNPOH]jJW.Dom ε>0δ>0xxP (|X + Y | > x) ≤ P |X| >+ P |Y | >,22(11.1)b!cWde(fhgiFjÀP] FHG=D&WNjmH\*QHWIUPIcmH\=KW]PNjVQX\*fcv_:V]j\=bk=D&dHOH_:V5WNj\*QHkHQHOQ n]j\*Njm=b.N^Jnx>0X YkJIP(|ξ − η| > δ) = P(|ξn − ξ − ξn + η| > δ) ≤≤ P(|ξn − ξ| > δ/2) + P(|ξn − η| > δ/2) < εmH\=K5W]PNjV vI\*l&pQVOHNPYI&JIGHIUPI.JIQJGHNPvI&W.D\*I]PlEFHIYDsLDJlHnFHI]PYI\*lYb\*NPW.DKuk=D&]^nJlzOHNPG=D&WNPOH]^JW.DzOHNsLD&WQHn]P0QJ9I&J n¾x%ÄSxTx»Ô¢HIYDsLDJlHnHkJIQsE]eVItm*QHRTI]^JQOHN]j\*Njm=bN^J9]eV.ILm*Qodξn −→ ξRTI]^Jl jPξn −→ ξ n → ∞ÁaCE -b.]^JlnQξn ≡ η ∼ Bi(1, 1/2) ξ = 1 ⇔ η = 0 ξ = 0 ⇔ η = 1¤kHNPWQm*OHIHn=kJIn=OHIdom |ξn − ξ| ≡ 1ξn −→ ξM5~_7 77¾ x%ÄSxTxË TIYDstDJlHnakJI Qs]eV.Itm*QHRTI]^JQ]j\*Njm=bN^J]eV.ILm*Qoξn −→ξîïðïRTI]^Jl jPξn −→ ξ n → ∞¾x%ÄSxTxÀÔÄSIYDsLDJlzOHN^G=D&WNPOH]^JWI)Ó j¾x%ÄSxTxÀ Ö *b]^JlQÄSIYDsLDJlPP [ @ ξn −→ ξnPaξn + bηn −→ aξ + bη a, b ∈ IR~P|ξn | −→ |ξ|Pξn ηn −→ ξη¾x%ÄSxTxÀ *b]^Jl [ @ ~ηn −→ ηξn −→îï ðï ξnQaξn + bηn −→îï ðï aξ + bη a, b ∈ IR|ξn | −→îï ðï |ξ|~ηn −→î.ï ð.ï η~ξn ηn −→îï ðï ξη¾x%ÄSxTxÀ =IYDsLDJln=kJI "NP]j\*Q[ "NP]j\*QPξn −→ a 6= 0ξn −→îï ðï a 6= 0nHJInHJI~P1/ξn −→ 1/a1/ξn −→îï ðï 1/aÄSIYDsLDJls jjt ©:®¬{²¿Ù Î Úe¬®v¤:ÚÙ9z«jàqÙ w Æß.ÞàHv² Î ª« w ®ß&®H[¾ x%ÄSxax; ÄSIYDsLDJlHn=kJIzNP]j\*QnHJIPξn −→ ξξn =⇒ ξ¾x%ÄSxaxÑ Tça\*Njm=bN^J \*QgQsy]eVItm*QHRTI]^JQ]rV.Itm*QHRTI]^Jldξn −→ ξξn −wdξ −→ 0¾x%ÄSxaxË @ aIYDsLDJlHnakJIJIUrmD QgJI\*lYI JIUemDnYIUemDξn =⇒ aPξn −→ aQ*IYDsLDJlHn=kJI¾x%ÄSxax; t£H=b]^JldP[ [ dξn + ηn −→ ξPξn ηn −→ 0~ξn −→ ξηn −→ 0QFHGHQ¾x%ÄSxax ¢ÄSIYDsLDJlHnkJIuNP]j\*QPPξn − an −→ 0 ξn − bn −→ 0nUem*NQ{Çm*WN%FHI]j\*Njm*IW.DJNj\*lOHI]^J.Q9WN^iNP]^JWNPO=OH_:Vn→∞kHQH]PNj\n=JI a1 , a2 , .
. . FHb1GH,Q b2 , . . . an − bn −→ 0n→∞¾x%ÄSxaxÇ Lb]^JlFHGHQÄSIYDsLDJlHnkJI 2 PP(ξn − ξ)2 −→ 0n→∞ξn −→2 FHGHQξn→∞¾x%ÄSxax GHQHWNP]^JQFHGHQHRTNPGnFHIYDst_-W.D&f[iQHdnkJIyQsz]eVItm*QHRTI]^JQWy]PGHNjm*OHNPR¯\*fcvIU^IuFHI\*ILM9QJNj\*lOHIUPIFHIG.Km*YDyOHN5]j\*Njm=bN^J)]eV.ILm*QHRTI]^JlFHIkJQO=D&WNPGHOHIN&¾x%ÄSxaxÅ Ö GHQHWNP]^JQ§FHGHQHRTNPGgFHI&]j\*Njm*IW.DJNj\*lOHI]^J.Qg]j\=bk=D&dHOH_:V§WNjo\*QHkHQHOn]eV.ILm=KiNPdH]rKFHIkJQ`O=D&WNPGHOHIN&nQ J&D&YIdnkJIOHQHYD&YDKNPNzFHILm*FHI]j\*Nm*IoW.DJNj\*lOHI]^Jl9OHNE]eV.ILm*QJ]rKWz]PGHNjm*OHNPR¸FHIG.Km*YDr>0¾x%ÄSxaxû LÄSIYDsLDJlHnkJI+]j\D&v.DK]eV.ILm*QHRTI]^Jl+ZbOHYHHQHdG=D&]PFHGHNjm*Njo\*NPOHQK§YgOHNPFHGHN^GH_-WOHId¨ZbOHYHHQHQgG=D&]PFHGHNjm*Nj\*NPOHQHKg|PYHWQHW.D\*NPOJO=D FHI&JIkHNPkHOHId]eV.ILm*QHRTI]^JQQuG=D&WOHIRTNPGHOHIdu]eV.ILm*QHRTI]^JQO=DIR¾x%ÄSxax & -b]^Jl{íFHI]j\*Njm*IW.DJNj\*lOHI]^Jl¸OHN^sLD&WQH]PQHRT_:Vξ1 , ξ 2 , .
. .]j\=bk=D&dHOH_:V WNj\*QHkHQHOnFHGHQHkHNPRFHGHQÄSIYDsLDJlHnkJI QHRTNPN^JPξn −→ ξn→∞ξW_-GHILM5m*NPOHOHINEG=D&]PFHGHNjm*Nj\*NPOHQ=N&[ñ÷4 ó)( ä ã áã§õø¬ç«ª ã ÷ Ûã ßäãK ªãÛßäèè B Ûã ä߸ö¼÷ çß»Ê`÷ ã¨Ðò Ìø ùûúýüËþ¿ÿ b]^Jl+sLDmD&OHI+OHNPYI&JIGHIN"WNPGHILKJOHI]^JOHIN'FHGHI]^JG=D&OH]^JWIQ`FHIo(Ω, F, P)]j\*Njm*IW.DJNj\*lOHI]^J.l]j\=bk=D&dHOH_:VWNj\*QHkHQHOO=DOHNPR-ξ1 , ξ 2 , . .
.¤EÁ Ä ¦Ã aC Ñ VH"Ià2A\)+\EI"4IBEH.[ξ , ξ , ...><><1!K!$&)%MU^_B'!><*1'!¾AB=*)+><+!$& 7\'GIE 7"\'Z 7+"$0'!A>U-=E^'< 7'¾^J98F;+'!=)* !'</ ^< 7IE'k"Rqa'<%`><U'^!-\ CÒ Ä*& Ó C5-B'QA' n → ∞1Pni=1 ξiS [−nPni=12EξiP−→ 0.¾ x axCE ¥ %GHNPvIW.DOHQHN5]^biNP]^J.WIW.D&OHQK RDJNPRaDJQHkHNt]PYHQVIM9QomD&OHQHd b)]j\=bk=D&dHOH_:VWNj\*QHkHQHO nnKHW\=KHN^J]rK]^biNP]^J.WNPOHOH_-R%nFHIoξi i = 1, 2, . . .]PYI\*lYbyW9FHGHIJQHWOHIR]j\=bk=D&NSIzW_-FHI\*OHNPOHQHQsLD&YIO=D5vI\*l&pQVkHQH]PNj\)WIIv&iNOHNERTIMN^JzQm=JQuGHNPkHQ¤EÁ Ä ¦Ã aC Ñ VH"Ià2A\)+\EI"4IBEH.[[ξ , ξ , ...><><1!S!$&)U%M^LB'!*><1'!¾AB=*)++><!$&\ 7'GIE 7"'K\ 7+$0"A'!`>U-<^=E'< 7'¾^n9H8F''!)!!!'<$0^<1 7';+Rq=a* `/ U^!IE.Ck"'<%M><'-B Ò ° Ä*& Ó C5-B'MA' n → ∞1Pni=1 ξi−nPni=12Eξi−→îï ðï 0.@VH"IJ2A*\+)*.[ξ , ξ , ...+\IE"ID;++''< 7$&%G*)'!!=lA)+B!!$&%~\HE><1!!$&%GB'!><'!UC ¤Á ux1 ¾ WZEIGHRTNNPv_pNPW.D. j @12s j ¢ t,+ x «¬« w ³v²c«z«PÚ x «D¤0²z«PÚ x ®H®ª« x Úy¥ztÚÙ9z«jàAA`'!<>U^\ Eξ^¾;+==a* ' /DξIE=k"'<σ%M><<'∞-\vRC5½vPD)-)"UI,^[)U^[*1¾6 A/ BP# )++BAIE' "'Q+$0.ε>0n→∞t£112P! n ξ i=1 i− a ≥ ε −→ 0.P n@¾ ¼WqZEIGHRTN,ïQHOHkHQHO=D. jl\'ø2A\)*.[[ξ , ξ , ...+¾=*BIE><!$& 7"'nIG 7D;++\' 7'<" 7'!$&>U%Q-*=E)'<'! 7!'F=98F'!)!A'<^< 7)+\'UC!!O$&%O)UB^]HE*><1!9!14$&A%Q\\)+'!><+'! .BIE"'Q+$0A`<^^S;+=* / IEk"'<%M><'-B¥RÑ2AB+)+.@¤Áuxû [ @ ½E¾ ÂI\*RTIUPIGHIW.D. jYVH"Iξ , ξ , ..."UBB^4IEIE+$0A""In'L!-D;+'</^Ü%+'*H)'<'<' 7S$&!%:'L*)+P#)¾'!!";+==*!*[></ [AIEHEk"?'<-)+%ÜB><!'!-+$&\4%_!')B(U^KHQHE><\1!HE!*><1Ü$&%:1!A*\\1'!)+><\'!+'!..R><'!!$ ξ =*><P# 7\ 7"'!>U-=P#98F'!)!'<^RCNP]PRTI&JG.KO=D"JIHnkJIyJNPIGHNPRT_ [ ^o [ @ QHRTNPf[JIkHNPOHluFHItV.IMzbf;ZEIGHR¦b.o\*QHGHIWYbn.RTN^M5m=bFbv\*QHYD&HQKHRTQ9FHNPGHWIdzQ5FHI]j\*Njm*OHNPd9JNPIGHNPR`FHGHIpX\*ISOHNPRaD\*IWGHNPRTNPOHQ`QFHILKHWQ\*I]PlvI\*l&pINzYI\*QHkHNP]^JWI+OHIW_:VRTN^JItm*IW+Qs^bkHNPOHQK]^bRTR]j\=bk=D&dHOH_:V`WNj\*QHkHQHONPIGHNPRaD [ 9v_:\DuIFbv\*QHYIW.D&O=DÃzNPv_pNPW_-RW Ö .`UPItm=bW§G=D&vI&JN¨NPv_pNPW¯ Ö . jnDb]PQ\*NPOHOH_-d¯sLD&YIOêvI\*l&pQVkHQo ¤Á uxË 11221]PNj\¸xSTCTI\*RTIUPIGHIW.D§vN^s"m*IYDsLDJNj\*l]^JW.D. bWQm*Nj\¸]PWN^J)\*Qpl W)G=D&vI&JN±²´³z±&µ±¶e±&·L @@ j¤EÁ Ä ¦Ã aCE ¸ @ !7$&~%r½ BC5-'!B><'!'M Aξ\,ξ)+, +.
..[B+U$0IEaA`<U"^^!I[\Ch><^3HE><1!)!vU$Y^[%QA9BIEB!'!*><)+^r'!+ A\)+BIEIE!#*^r'[\-`HE>< 71! .Ò Ó12Pni=1 ξiq−EPnPni=1 ξiB>< 1!/ "*1Q%*B)'!'!><'!^S9AR ' n → ∞ =("!)#"V` 79IEA)+\!*1¾\HE.@¤Áuxë [ £Ç.-EÅmH\=KOHNjsLD&WQH]PQHRT_:VÇItm*QHO=D&YIWI¸G=D&]PFHGHNjm*Nj\*NPOHOH_V]j\=bk=D&dHOH_:V¸WNj\*QHkHQHO* jVH"I 7$&%L*)'!!=R½vAPD))+\)U^n!!$&ξ%*1],\ξAHE,><B..1!.!)+2Ð$&+%LA\BB'!IE><)+'!+U"CA'4\'!+IE$0>U\A "Eξ<I^=D;+a^]+C '0'!!<..Dξ = σ < ∞9IE!*^MA)+BIE!*^Q-` 7C5-"I,A'1Di=1 ξi2112n→∞PPn!x− na1 Z − u2√e 2 du.< x =⇒ Φ(x) = √2π −∞nσ 2i=1 ξi(12.1)S [ [S [b!cWde(fhgiFj+Í¢[¾ x axCE :IRTOHIUPQV9bkHNPvOHQHYDV"FHIJN^IGHQHQyWNPGHILKJOHI]^JNPdyJGHNjovIW.D&OHQHNOHNPW_-GHILM5m*NPOHOHI]^JQ ]j\D&UjD&NPRT_:VW'ZEIGHR¦bt\*QHGHIWYNJNPIGHNPRT_} £"OHNbYDst_-WD&N^J.]rKKHWOHIHnItm*O=D&YIHnIkHNPWQm*OHIHnkξJi I+b.]j\*IW&QHN 2KHW\=KHN^J]rK]^biNjoσ > 0]^JWNPOHOH_-R-¾x axCE ¼ @ ça\D&v.DK¸]rV.Itm*QHRTI]^Jlg|PYHWQHW.D\*NPOJO=D]eV.ILm*QHRTI]^JQFHIG=D&]PFHGHNjm*Nj\*NPO=QHfSnHJINP]^JlWJNPIGHNPRTN £SbJWNPG.M5mD&N^J]Ky]eVItm*QHRTI]^Jl5ZbOHYHHQHdG=D&]PFHGHNjm*Nj\*NPO=QKqHNPOJGHQHGHIW.D&OHOH_:VqQ+OHIGHRTQHGHIW.D&OHOH_:V)]^bRTR]j\=bk=D&dHOH_:V)WNj\*QoWYDM5m*IdgJIkHYNOHNPFHGHNPGH_-WOHI]^J.QFHIokHQHO¸Y¨ZbOHYHHQHQ¸G=D&]PFHGHNjm*Nj\*NPOHQKΦ(x)]j\*Njm*OHNPd¤[m*O=D&YIHn=ZbOHYHHQKKHW\=KHN^J]KD&v]PI\*f[JOHIzOHNPFHGHNPGH_-WOHIdn]j\*Njm*IoΦ(x)W.DJNj\*lOHI]j\D&v.DK9]eV.ILm*QHRTI]^JlW'Ó T|PYHW&QHW.D\*NPOJO=DFHI&JIkHNPkHOHId"]rV.Itm*QHRTI]^JQmH\=K"\*fcvIUPIm*NPdH]^JWQJN^\*lOHIUPI = :I\*NPNJIUPIHnHRTIM9OHIFHIYDsLDJlHnHkJI5FHI]PYI\*loYb§YD&Ym*IFHGHNjm*Nj\*lOH_-N&nJD&Y¨Q¨xFHGHNjm*Nj\*lO=DK¨ZbOHYHHQHQ¨KHW\=KHf[J]rK¨ZbOHYHHQKHRTQG=D&]PFHGHNjm*Nj\*NPO=QKn]j\D&v.DK)]eV.ILm*QHRTI]^JluW Ó %|PYHWQHW.D\*NPOJO=DyG=D&WOHIRTNPGHOHId)FHI]rV.Itm*QHRTI]^JQx ∈ IR¾x%ÄSxTxÀ &ÄSIYDsLDJl5JN^IGHNPR¦b ÁaCE -[¤v_-kHOH_-R}RTN^JItm*IRäFHGHQm*IYDsLDJNj\*l]^JWNqsLD&YIO=DvI\*l&pQVkHQH]PNj\¥KHW\=KHN^J]rKÇFHGHQHRTNPOHNPOHQHN§OHNPG=D&WNPOH]^JW.DêNPv_pNPW.D¿ Ö . j¾D&RTN^JQHR-nkJInH]j\*Njm*IW.DJNj\*lOHIPn[[om i=1 ξi /nEFHGHQ[[[=aP!P n ξD ni=1 ξiσ2 i=1 iP = 2 −→ 0− a ≥ ε ≤nn2 ε 2nε¾x%ÄSxTxê GHIWItm=KJ]rKOHN^sLD&WQH]PQHRT_-NQH]PFH_JD&OHQK5 :NPGHObL\H\*Q]WNPGHILKJoOHI]^Jlfb]PFHNjVD .b.]^JlX]j\=bk=D&dHO=DK9WNj\*QHkHQHO=DFHGHQHOHQHRaD&N^JXstO=DkHNPOHQHN nNP]j\*QoNyQoNupQH]PFH_JD&OHQK¸sLD&YIOHkHQ\*QH]Plb]PFHNjVξIi R;QWFHGHI&JQHWOHIR;]j1\=bk=D&N&i0:_-FHI\*(iOKH+N^J1)]Ky\*Q'mH\=KuFHI]j\*Njm*IWDJNj\*lOHI]^J.QsLD&YIOuvI\*l&pQVykHQH]PNj\wξ1 , ξ 2 , .
. .ÁaCE -b]^Jly]j\=bk=D&dHO=DK)WNj\*QHkHQHO=D]PII&JWN^J]^JW&b.N^J oR¦buQH]PFH_JDoηiOHQHf; :NPGHObt\H\*QIUemD¾D&RTN^JQHR-ni kJI']j\=bk=D&do{ξi = 1} = {ηi = 1, ηi+1 = 1}OH_-NzWNj\*QHkHQHOH_QHRTNPf[J+G=D&]^FHGHNjm*Nj\*NtOHQHNnOHIuOHN5KHW\=KHf[J]rKOHN^sLD&WQoξiBi(1, p2 )]PQHRT_-RTQCED&dm*NPR*RTNPNPRPn[[n→∞om DEnXi=1ξi!2= E= EXi=jnXi=1i=1 ξiξi ·nXj=1(ηi ηi+1 )2 + 2ξj = E Xj=i+1Xξi ξj +i=j2ηi ηi+1ηi+2 +Xξi ξj +|j−i|=1X|j−i|>1X|j−i|>1ξi ξj =ηi ηi+1 ηj ηj+1 == np2 + 2(n − 1)p3 + (n2 − n − 2(n − 1))p4 .s j ¢ t,+ x «¬« w ³v²c«z«PÚ x «D¤0²z«PÚ x ®H®ª« x Úy¥ztÚÙ9z«jàL¦D&YHQHRgIvG=DstIR-nDnXξi = Ei=1nXi=1ξi!2− EnXi=1ξi!2= np2 + 2(n − 1)p3 + (2 − 3n)p4 . GHQHRTNPOHQHW§OHNPG=DWNPOH]^JWI1 Ö . zY]j\=bk=D&dHOHIdWNj\*QHkHQHOHN Pnn:FHI\=bk=D&NPR-nξi /ni=1kJI5sLD&YIOuvI\*l&pQVykHQH]PNj\)W_-FHI\*OKHN^J]rK¾x%ÄSxaxÅ @ ¦b.]jJl{ýOHN^sLD&WQH]PQHRT_-Ny]j\=bk=D&dHOH_-NuWNj\*QHkHQHOH_nξ1 , ξ 2 , .
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