А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 18
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. .FHGHQHkHNPR[√n,1;2n1ξn = 0,1− ;n√1 − n,.2n:_-FHI\*OKHN^J]rKu\*Q"mH\=KuFHI]j\*Njm*IW.DJNj\*lOHI]^J.QLs D&YIOuvI\*l&pQVykHQH]PNj\wξ1 , ξ 2 , . . .om Q PGHQHRTNjoÁaCE -¾DRaNjJQHR%n*kJI PnE i=1 ξi /n = 0 D ni=1 ξi /n = 1/nOHNPOHQHNXOHNPG=D&W&NPOH]^J.WD) Ö sLD&WNPGpD&N^J"GHN^pNPOHQHNEsLDmDkHQ¾x%ÄSxaxÅ £H¦b.]jJl{ýOHN^sLD&WQH]PQHRT_-Ny]j\=bk=D&dHOH_-NuWNj\*QHkHQHOH_nξ1 , ξ 2 , . . .FHGHQHkHNPRE:_-FHI\*OKHN^J]rK¿\*Q1mH\=K¿FHI]j\*Njm*IW.DJNj\*lOHI]^J.QP(ξn = ±2n ) = 1/2sLD&YI&OuvI\*l&pQVykHQH]PNj\wξ1 , ξ 2 , . . .ÁaCE -c¾D&RTN^JQHR-nkJI Pn-GHQHRTNPOHNPOHQHNqOHNPG=D&WNPOH]^JW.DE i=1 ξi /n = 0 Ö . :Y'mDOHOHIduFHI]j\*Njm*IW.DJNj\*lOHI]^J.QymD&N^J[om P!PnPnPi n ξD ni=1 ξi i=1 ii=1 2i=1 ξi P =−→ ∞−E≥ε ≤nn n2 ε 2n2 ε 2HF GHQH¦D&YId'GHN^sPbt\*lJDJzKHW\=KHN^J]rKyFHNPGHW_-R-n=OHIOHNW]PNPUemDWNPGHOH_-R-n=FHGHQon→∞stO=D&YIRJIUPIHnkJI'sLD&YIOqvI\*l&pQV)kHQH]PNj\`OHNW_-FHI\*OKHN^J]rKb.]^JlIoε<1]PYI\*lYbQHRTNPNPRP ξ1 + .
. . + ξn−2 ≥ 2 − 2n−1 = 1,PP!! n ξ n ξ i=1 i i=1 i nn−1P ≥ε ≥P ≥ ε, ξn = 2 , ξn−1 = 2=n n P! n ξ i=1 i ≥ ε ξn = 2n , ξn−1 = 2n−1 P ξn = 2n , ξn−1 = 2n−1 == P n 1= P ξn = 2n , ξn−1 = 2n−1 = .4b!cWde(fhgiFj+ͦD&YHQHR¯IvG=DstIR-nWymD&OHOHIR¯]j\=bk=D&NzsLD&YIOvI\*l&pQV kHQH]PNj\`m*NPdH]^JWQJN^\*lOHI+OHNW_-FHI\*OKHN^J]rK¾x%ÄSxTx Ô¢.b]^Jl{¥OHN^sLD&WQH]PQHRT_-N[Itm*QHO=D&YI&WIEG=D&]PFHGHNjm*Nj\*NPOoξ1 , ξ 2 , . . .OH_-N]j\=bk=D&dHOH_-NWNj\*QHkHQHOH_nFHGHQHkHNPR.ÄSIYDsLDJlHnkJIXmH\=Kz\*fcv_:V0 < Dξ1 < ∞YIOHNPkHOH_:V[a, b ∈ IRlim P a ≤omaCE-¤vI&stO=DLkHQHRn→∞ÁnXQσ 2 = Dξ1nXlim P a ≤n→∞[ξi ≤ b = 0.i=1m = Eξ1a − mn≤= lim P √n→∞σ2n!i=1Pn=RTNPNPR!ξi ≤ b =!− mnb − mn√≤ √.σ2nσ2ni=1 ξi[ GHQHRTNPOHNPOHQHNXsLD&RTNPk=D&OHQK @ I5G=D&WOHIRTNPGHOHId]eV.Itm*QHRTI]^JQ+W5JNPIGHNPRTN9 £m*IoYDst_-WD&N^JzJGHNPv&b.NPRTINEbJWNPGMm*NPOHQHNM5~_7 77[[¾x%ÄSxTxÀ *IYDsLDJlz]^FHG=D&WNjmH\*QHWI]^Jl9sLD&RTNPk=D&OHQK @ ¾x%ÄSxTxÀ Ô ÄSIYDsLDJl5JN^IGHNPR¦b £H¾x%ÄSxTxÑ Ö ¦b]^Jl{ýOHN^sLD&WQH]PQHRT_-Nu]j\=bk=D&dHOH_-NuWN\*QHkHQHOH_nξ1 , ξ 2 , .
. .FHGHQHkHNPR[[[1;2n21ξn = 0,1 − 2;n1 −n,.2n2:_-FHI\*OKHN^J]Ky\*Q'mH\=KuFHI]j\*Njm*IWDJNj\*lOHI]^J.QsLD&YIOuvI\*l&pQVykHQH]PNj\wξ1 , ξ 2 , . . .{ýOHN^sLD&WQH]PQHRT_-Nu]j\=bk=D&dHOH_-NuWN\*QHkHQHOH_n¾x%ÄSxTxÑ ¦b]^Jlξ1 , ξ 2 , . . .n,[FHGHQHkHNPRn2 ,ξn = 0,−2n ,2−(2n+1) ;1 − 2−2n ;2−(2n+1) .:_-FHI\*OKHN^J]Ky\*Q'mH\=KuFHI]j\*Njm*IWDJNj\*lOHI]^J.QsLD&YIOuvI\*l&pQVykHQH]PNj\wξ1 , ξ 2 , . .
.¾x%ÄSxTxû b]^Jl{ÑYI\*QHkHNP]^JWI+b]PFHNjV.IW)WOHN^sLD&WQH]PQHRT_:V`QH]joµnnFH_J&D&OHQKV :NPGHObt\H\*Q+]EWNPGHILKJOHI]^JlfÇb]PFHNjVD ÄSIYDstDJlHn=kJIpµn /n −→îï ðï p[ Ö[s j ¢ t,+ x «¬« w ³v²c«z«PÚ x «D¤0²z«PÚ x ®H®ª« x Úy¥ztÚÙ9z«jྠx%ÄSxaxÑ &Tb]^Jl{ξ1 , ξ 2 , . . .\*NPOHOH_-NX]j\=bk=D&dHOH_-NSWNj\*QHkHQHOH_n*FHGHQHkHNPR[ [OHN^sLD&WQH]^QHRa_NuILm*QHO=D&YIWIqG=D&]PFHGHNjm*No*:_-kHQH]j\*QJl0 < Dξ1 < ∞lim P(ξ1 + . . . + ξn < x).n→∞¾x%ÄSxaxä b]^Jl{ OHN^sLD&WQH]PQHRT_-N]j\=bk=D&dHOH_-NWNj\*QHkHQHOH_nξ1 , ξ 2 , .
. .FHGHQHkHNPR n,ξn = 0,−n,1/4;1/2;1/4.:_-FHI\*OKHN^J]rK9\*QzmH\=K9FHI]j\*Njm*IW.DJNj\*lOHI]^J.QsLD&YIO9vI\*l&pQV9kHQH]PNj\wξ1 , ξ 2 , . . .¾x%ÄSxaxä @ b]^Jl{ OHN^sLD&WQH]PQHRT_-N]j\=bk=D&dHOH_-NWNj\*QHkHQHOH_nξ1 , ξ 2 , . . .√FHGHQHkHNPR%:_-FHI\*OKHN^J]Kê\*QmH\=K¯FHI]j\*Njm*IW.DJNj\*lOHI]^JQP(ξn = ± n) = 1/2sLD&YI&OuvI\*l&pQVykHQH]PNj\wξ1 , ξ 2 , . . .¾x%ÄSxax¸ t£HjÄSIYDsLDJl-sLD&YIOvI\*l&pQVEkHQH]PNj\W-ZEIGHRTNa :NPGHObL\H\*Q"WFHNPGo{ÀFHI]j\*Njm*IoW_-NIFbv\*QHYIW.D&OHOH_-dyWYHOHQHUPN9. /0 P¶ H2± 1H²´² @ e jhFb.]jJlξ1 , ξ 2 , .
. .W.DJNj\*lOHI]^JlOHN^sLD&WQH]PQHRT_:V9ILm*QHO=D&YIWIXG=D&]PFHGHNjm*Nj\*NPOHO=_:V"]j\=bk=D&dHOH_:V"WNj\*QHkHQHOnnI&UrmD"mH\=K |^JId`FHI]j\*Njm*IW.DJNj\*lOHI]^J.Q`W_:oFHGHQHkHNPRξ1 ∼ Bi(1, p) p ∈ (0, 1)FHI\*OKHN^J]rKusLD&YIOuvI\*l&pQVykHQH]PNj\¾x%ÄSxax¸ ¢jÄSIYDsLDJl-sLD&YIOvI\*l&pQVEkHQH]PNj\W-ZEIGHRTNab.D&]P]PIO=DSWFHN^GoW_-NzIFbv\*QHYIW.D&OHOH_-d`WYHOHQHU^N). 3± e± g Ö@ e jhFb]^Jl{ÅFHI]j\*Njm*Ioξ1 , ξ 2 , . . .W.DJNj\*lOHI]^JlOHN^sLD&WQH]PQHRT_:V]j\=bk=D&dHOH_:VWNj\*QHkHQHO¨J&D&YHQVn:kJInξi ∼ Bi(1, pi )nIUemD9mH\=Kq|^JId FHI]j\*Njm*IW.DJNj\*lOHI]^J.QW_-FHI\*OKHN^J]rKpi ∈ (0, 1) i = 1, 2, .
. .sLD&YIOuvI\*l&pQVykHQH]PNj\¾x%ÄSxax L-b]^Jl{íFHI]j\*Njm*IW.DJNj\*lOHI]^Jl¸OHN^sLD&WQH]PQHRT_:Vξ1 , ξ 2 , . . .Itm*QHO=D&YIWIG=D&]PFHGHNjm*Nj\*NPOHOH_V¸]j\=bk=D&dHOH_V¸WNj\*QHkHQHOTÄSIYDsLDJlHnakJI`NP]j\*Q¨FHGHQ[[`ô ó[óõò`ò ô[n→∞ξ1 + . . . + ξ n−→îï ðï c,nUem*N {}OHNPYI&JIGHINXWN^iNP]^JWNPO=OHINXkHQH]j\*IHn=JIQcE|ξ1 | < ∞ Eξ1 = c[¾ x%ÄSxax -b]^Jl{íFHI]j\*Njm*IW.DJNj\*lOHI]^Jl¸OHN^sLD&WQH]PQHRT_:Vξ1 , ξ 2 , . . .Itm*QHO=D&YIWI'G=D&]PFHGHNjm*Nj\*NPOHOH_VqHNPOJGHQHGHIW.D&OHOH_:V ]j\=bk=D&dHOH_:VqWNj\*QHkHQHOq]YIOHNPkon=NP]j\*QOH_-RTQ'm*QH]PFHNPGH]PQKHRTQCED&dJQDξ1lim Pn→∞[Pni=1 ξi√n!1>1 = .3¾ x%ÄSxax Ö -b]^Jl{íFHI]j\*Njm*IW.DJNj\*lOHI]^Jl¸OHN^sLD&WQH]PQHRT_:Vξ1 , ξ 2 , . . .]j\=bk=D&dHOH_:VWNj\*QHkHQHOnQHRTNPf[iQVG=D&WOHIRTNPGHOHIN9O=DI&JGHN^stYNG=D&]PFHGHNjm*Nj\*Njo[0, 1]b!cWde(fhgiFj+ÍLHO QHN&CED&dJQ9FHI]j\*Njm*IW.DJNj\*lOHI]^JlWN^iNP]^JWNPO=OH_:V"kHQH]PN\nbLm*IW\*N^JWIoa1 , a 2 , .
. .GKHf[iQV'b.]j\*IWQHflim PUrm*N0≤p≤1n→∞[nXi=1√!ξi ≤ an n = p,¾ x%ÄSxTx L-b]^Jl{íFHI]j\*Njm*IW.DJNj\*lOHI]^J.l¨OHN^sLD&WQH]PQHRT_:Vξ1 , ξ 2 , . . .ILm*QHO=D&YIWIG=D&]PFHGHNjm*Nj\*NPOHO=_:VyOHNPI&JGHQH=DJN^\*lOH_:Vu]j\=bk=DdHOH_:VyWNj\*QHkHQHOnHFHGHQHkHNPR=b.]^JlEξ1 = aηn =ξ1 + . . .
+ ξ n.nÄSIYDstDJlHnkJIqFHI]j\*Njm*IWDJNj\*lOHI]^J.lq]j\=bk=D&dHOH_:V§WNj\*QHkHQHO√ζn = n η1 · . . . · η n]rV.Itm*QJ]KFHIkJQuO=D&WNPGHOHINEY a¾x%ÄSxTx & CED&dJQqFHGHQHv\*QMNPOHOHINstO=DLkHNPOHQHNSmH\=KqWNPGHILKJOHI]^JQqJIUPIHnkJIkHQH]j\*ISºb.]PFHNjV.IWW]eVNPRTNOHN^sLDWQH]PQHRT_:VyQH]PFH_JD&OHQHd :NPGHObt\H\*QSnn = 100]WNPGHILKJOHI]^Jlf¨ºb.]PFHNjVD\*N^M9QJWFHGHNjm*Nj\DVI&Jm*IQI&Jm*I356547GHQYD&YHQV stO=DLkHNPOHQKV pWNP=GHIL0.5KJOHI]^JlJIUPIHnkJInv&bLm*53N^Jn0.35 ≤ Sn /n ≤ 0.65vI\*l&pNw[[0.998ä èæõ ß å Ûã çèã54NÊæä æ¿ÌÅ÷ ãÛäèËÊ`÷ èÑÌèß»Êhçß B «ç ª Ê ÎÛæ¿Ì ßçß B ä æ¥ÊÈäã áã§õ ãÛçèûö(fξ (x)PξP(ξ = k){ξ ár=â ãrâ a{1 k=aa0Cnk pk (1 − p)n−k ,n ∈ IN, p ∈ (0, 1),k = 0, .
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