А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 16
Текст из файла (страница 16)
. . , n¾x%ÄSxax¿ b.]^Jl Q {;OHN^sLD&WQH]PQHRT_-N]j\=bk=DdHOH_-NWNj\*QHkHQHOH_nFHGHQokHNPR QHRTNPN^JFHI&YDsLDJNj\*lOHINcξ G=D&]PηFHGHNjm*Nj\*NPOHQ=N][F=D&G=D&RTN^JGHIR nD G=D&WOHIRTNPGHOHIG=D&]PFHξGHNjm*Nj\*NPO=DzO=D5I&JGHN^stYN*CED&dJQuF\*I&JOHI]^Jl9G=D&]PFHGHNjm*1Nj\*NPO=ηQK[0, 1]ξ+η¾x%ÄSxaxê @ b]^Jl Q {1OHN^sLD&WQH]PQHRT_-N]j\=bk=D&dHOH_-N-WNj\*QHkHQHOH_`J&DYHQHN&nkJIS]^bRTRaDQHRTNPN^JSW_-GHξIMm*ηNPOHOHIN[G=D&]PFHGHNjm*Nj\*NPOHQ=N&.ÄSIYDsLDJlHnkJIXYDM5mDKξ +ηQs']j\=bk=D&dHOH_:V§WNj\*QHkHQHOQQHRTNPN^JqW_-GHIM5m*N^OHOHIN'G=D&]PFHGHNjm*Nj\*NPOHQHN&uIM9OHI\*Qu|^JI5bJWNPG.M5mDJlHn*NP]j\*Q ξ Q η OHNKHW\=KHf[J]rKuOHN^sLD&WQH]PQHRT_-RTQ*wξ η¾x%ÄSxax¿ t£Hb.]^Jl Q {;OHN^sLD&WQH]PQHRT_-N]j\=bk=DdHOH_-NWNj\*QHkHQHOH_nFHGHQokHNPR1]j\=bk=D&dHOH_-N"WNj\*QHkHQHOH_ ξ η QItm*QHO=D&YIWIuG=D&]PFHGHNjm*Nj\*NPOH_E¦CED&dJQG=D&]joξ+η ξFHGHNjm*Nj\*NPOHQHNS]j\=bk=D&dHOHIdWNj\*QHkHQHOH_ η¾x%ÄSxaxä ¢b]^Jl { ]j\=bk=D&dHO=DK+WNj\*QHkHQHO=D"]SZbOHYHHQHNPd)G=D&]PFHGHNjm*Njo\*NPOHQKCED&dJQ ZbOHYHHQHξ f}G=D&]PFHGHNjm*Nj\*NPOHQK]j\=bk=D&dHOHI&UPI+WNPYJIG=DF2 (x, y) F (x)(ξ, ξ)¾x%ÄSxaxä Lb]^Jl { ]j\=bk=D&dHO=DK+WNj\*QHkHQHO=D"]SZbOHYHHQHNPd)G=D&]PFHGHNjm*Njo\*NPOHQKCED&dJQ ZbOHYHHQHξ f}G=D&]PFHGHNjm*Nj\*NPOHQK]j\=bk=D&dHOHI&UPI+WNPYJIG=DF (x)F2 (x, y)(ξ, |ξ|)¾x%ÄSxax b.]^Jl {¥]j\=bk=D&dHO=DKzWNj\*QHkHQHO=Dn.QHRTNPf[iDKzG=D&WOHIRTNPGHOHINO=D%I&JGHN^stYNG=D&]PFHGHNjm*Nj\*ξNPOHQHN&CED&dJQSG=D&]PFHGHNjm*Nj\*NPO=QHN]\=bk=D&dHOHIdWNj\*QHkHQHOH_n [0, 1]−1 ²−λξ λ>0¾x%ÄSxax» Ö b]^Jl {äOHNPW_-GHILM5m*NPOHO=DKy]j\=bk=DdHO=DKuW&Nj\*QHkHQHO=D]]PQHRoRTN^JGHQHkHOH_-R¯G=D&]PFHGHNjm*Nj\*NPOHQHNtξRêQ)YIOHNPkHOHIdm*QH]PFHNPGH]PQHNtdCEDdJQqYI|PZEZEQHHQHNPOJYIGHGHNj\=KHHQHQ]j\=bk=D&dHOH_:VWNj\*QHkHQHO Qξ |ξ|¾x%ÄSxaxä L*b.]^Jl {Â]j\=bk=D&dHO=DK+WNj\*QHkHQHO=DnFHGHQHkHNPRnξP(ξ > 0) = αnn=CED&dJQ è ±é´µP(ξ < 0) = β Eξ = a E|ξ| = b(ξ,ξ)¾x%ÄSxaxä & *I]^JGHIQJl"O=DzOHNPYI&JIGHIRWNPGHIKJOHI]^JOHIRFHGHI]^JG=D&OH]^J.WNm*WNOHNPYIGHGHNj\*QHGHIW.D&OHOH_-NX]j\=bk=D&dHOH_-NXWNj\*QHkHQHOH_Q nHYI&JIGH_-NXOHN(Ω, F, P)ξ ηKHW\=KHf[J]rKuOHN^sLD&WQH]PQHRT_-RTQ[õô[ò`ó ôè ᯪ ÊGÊßâáè ß»Ê`÷ ã¨ÐÊõ«·¼Ï æ¿Ð竪 Êã õ èÑÏèçB §ñò¥ò»óA »ø 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_:VqWNj\*QHkHQHO O=D"OHNPR- S9v_:\*IyFHIoξ 1 , ξ2 , .
. .YDsLD&OHInkJISG=D&WNPOH]^ξ,JWIS]j\=bk=D&dHOH_:V"WNj\*QHkHQHO9RTIM9OHIEIFHGHNjm*Nj\=KJlG=Ds^\*QHkHOH_-RTQ]PFHI]PIv.D&RTQ¦D&YMN%FHIoG=DstOHIR¦b5RTIM9OHIXIFHGHNjm*Nj\=KJlQ"]eV.Itm*QHRTI]^Jl]j\=bk=D&dHOH_:VWNj\*QHkHQHOÁD&]P]PRTI&JGHQHR¸I]POHIWOH_-NXWQm*_êJD&YHQVu]eVItm*QHRTI]^JNPd¤EÁ Ä ¦Ã aC T\=~ BHE><HE><1!!1!$& %Q*1OBB'!><'!'!><Q'! %*)Cl'!-\a'M^QAA`*U><' ^!"'MC[!><BKA \Ò )++B^!B"IE""IEI6Jξ#),'!ξ!'!, !. .Ó .ξn→∞12 / -;+!>U!'9 ξ −→Rîï ðï ξ¤EÁĦÃaC [ a`U^!C[><KA\)++BIE"I-\\HE'M><)1!U!^[$&%:6 / BP#'! ><'!:%A*)'!' ^:AB^!""':=n\HE><1!*1:Bξ'!><, '!ξ, .
ξ. .CP {ω| ξn (ω) −→ ξ(ω)} = 1.n12n→∞ε>0P(|ξn − ξ| > ε) −→ 0. / -;+!>U!'9 ξ −→ ξ RG\BHE><¤E)1!Á!\$& r%QÄ=¦+\Ã)a'!><C'!"Q'! >U%-*=)D'!@ Ca^-B(`' UB^!)C[ >< 3=[AK`BUA^!HE)><\=E1!)+r*+1Ò ,BBB'!IE)><'!\ " CI -C\ξ-'\,rξ'Q=,A.1.' .Cr = 2Óξn→∞nP1 / -;+!>U!'9 ξ(r)n−→ ξRE|ξn − ξ|r −→ 0. 2s jjt ©:®¬{²¿Ù Î Úe¬®v¤:ÚÙ9z«jàqÙ w Æß.ÞàHv² Î ª« w ®ß&®H¤EÁĦÃaCE £Ha U^!CS><KAB)++\IE"I-BBHE']><A1!!' $&%n\'!><'!nA\%*))+'!+^BA"IE"AI)+-\**!'!6¼=B+HHE6><?1!'%4*1n0HE!\=Eξ'!!,><'!'!1Qξ, .
ξ...C∞A)+B!'<n^ F→ (x),%*)'!^=0HE!=E!'!'¾A)+B!'<^ F (x) F (x), . . .=E+8F)+*1M*><=VAB$+"'MAB)1R / -;+!>U!'9 ξ −→ ξ R¤EÁ Ħ Ãa CE Ô¢a U^!CS><KAB)++\IE"I/B*1MHE><1!A!$&B%Q$+\'!*1Q><'!QP¿B! '!/ >U!%**)1Q'!0HE!^Q=E=G!'!\' HE><1!A*1Q' B'!><'! ξ Cl-\'Mξ ),Uξ^~,.6. ..f (x)n→∞ Ö1ξ1ξ22ξdn12Ef (ξn ) −→ Ef (ξ). / -;+!>U!'9 ξ =⇒ ξ R¤EÁĦÃaCE& +a`U^!C¥><A\)++BIE"IV0HE!=E!'!1@./)AU^[)+B6 / !*'<1Q^ FA,FB$, +. . .91Q\ P¿<Ov'!%>U*)!'!*1O^40=HE!0=EHE!!'!=E' !'!'AA')+\!'<^ F C-B'n12f (x)Zf (x) dFn (x) −→Zn→∞f (x) dF (x). / -;+!>U!'9 F (x) =⇒ F (x) R¾xSaxCE¼ -¤kHNPWQm*OHInkJI]j\D&v.DK]eV.ILm*QHRTI]^Jl§ZbOHYHHQHdG=D&]joFHGHNjm*Nj\*NPOHQKYzZbOHYHHQHQzG=D&]PFHGHNjm*Nj\*NPOHQHK|PYHWQHW.D\*NPOJO=DX]j\D&vIdn..]eV.ILm*QHRTI]^JQF]j\=ξ1b,k=FD&dHξ2OH, _:.
VWNj\*QHkHQHOYu]j\=bk=D&dHFOHξI&dWNj\*QHkHQHOHN ξ1 , ξ 2 , . . .ξ¾x axCE ¯ =I]PYI\*lYb'W5IFHGHNjm*Nj\*NPOHQKV £Q Ô¢SGHNPkHlzQm*N^JzIG=D&]PFHGHNjm*Nj\*NPOHQKV]j\=bk=D&dHOH_:V WNj\*QHkHQHOnD'OHN5Iy]tD&RTQVq]j\=bk=D&dHOH_:VWNj\*QHkHQHO=DLVnJGHNPvIW.D&OHQHNXNjm*QHOH]^JWNPOHOHI]PJQ)v.DstI&WIUPIzWNPGHILKJOHI]^JOHIUPI"FHGHI]^JG=D&OH]^J.WDnO=D5YIoJIGHIR;IFHGHNjm*Nj\*NPOH_û]j\=bk=D&dHOH_-N+WNj\*QHkHQHOH_naKHW\=KHN^J]rK¸W mD&OHOHIRξ, ξ1 , ξ2 , . . .]j\=bk=D&NXOHNP]^biNP]^J.WNPOHO=_-R- ¤Á uxæ + JNPI&GHNPRaDOHNPFHGHNPGH_-WOHI]^J.Q* jS [/ *^%*)'< "IQAB.@Wq|)++'++9B!"IE!M"%'*)B'< HE><1!"!'n$&%LA\\'!)+><+'! ξB, IEξ,.".
.'K=¾-\*HE><1!*1L+HEB6'!?"><'<'!%L %!#ξ=E=E..B'A"'!' >U-=E'<%[0HE!=E=E+!8F'!1 )+*ϕ1Q(t),*><ϕ= (t), . . .R =%!#=EB'"'!>U-=*1S0HE!=E!'!'ϕ (t)n→∞t ∈ IR¾xSaxCEæ @ -¤v_-kHOHIHn:mH\=K¸m*IYDsLDJNj\*l]^JW.Dg]j\D&vId]eVItm*QHRTI]^JQFHI\*l&sPbf[J]rK"QHRTNPOHOHIXJNPI&GHNPRTId+ nFHI]PYI\*lYbm*IYDst_-W.DJlXFHI&JIkHNPkHObf¯]rV.Io1ξ12ξ2ξm*QHRTI]^JlyZbOHYHHQHdqUPIG=Ds^m*IyFHGHI&iN&nOHN^MNj\*Qq]eVItm*QHRTI]^JluQHOJNPUPG=D\*IWyI&JyW]PNjVOHNPFHGHNPGH_-WOH_:VIUPG=D&OHQHkHNPOHOH_:VZbOHYHHQHd¾x axCE ¯ £HÄE\=KuG=Ds^\*QHkHOH_:VuWQm*IWz]eV.Itm*QHRTI]^JNtd]j\=bk=D&dHOH_:VWNjo\*QHkHQHOQQVG=D&]PFHGHNjm*Nj\*NPOHQHd5]PFHG=D&WNjmH\*QHW.D[m*Q=D&UPG=D&RTRaDcQHRTF\*QHYD&HQHdn&FHGHQHWNjm*NPOoO=DKyO=D5GHQH]^bOHYNz Sb!cWde(fhgiFjÀП.Н.ξn −→ ξ ξn =⇒ ξFn =⇒ Fdξn −→ξϕn −→ ϕPξn −→ξ(r )ξn −→ ξÏd»¼±º}££¾ £C DyFHG=D&YJQHYN"QH]P]j\*Njm*IW.DJNj\=KRTIU^bJ+QHOJNPGHNP]PIW.DJl)G=Ds^\*QHkHOH_-N9WQm*_;]eV.IoEm*QHRTI]^JNPduWSsLD&WQH]PQHRTI]^JQyI&JG=D&]P]PRaDJGHQHW.D&NPRTId'sLDmDLkHQHCED&FHGHQHRTNPGnsLDk=D&]^J&bfFHGHNjm*]^J&D&W\=KHN^J]KOHNPWI&stRTILM9OH_-RÂW_-kHQH]j\*QJ.l¸G=D&]PFHGHNjm*Nj\*NtOHQHN`OHNPYI&JIGHId1]j\=b.ok=D&dHOHIdWNj\*QHkHQHOH_ W[JIkHOHI]^JQnmDMNNP]j\*QQstWNP]^JOH_gW]PNF=D&G=D&RTN^JGH_§RTILm*Nj\*Q¤[m*O=D&YIHnN^]j\*QFHGHQ|^ξJIR+btmD&N^J]rKFHIYDsLDJlHnkJIcOHNPQstWNP]^J.O=DKZbOHYHHQKG=D&]PFHGHNjom*Nj\*NPOHQK]j\D&vI9]^JGHNPRTQJ]K)YQstWN^]^JO=Id+ZbOHYHHQHQ+G=D&]PFHGHNjm*Nj\*NPOHQHKFHGHQFξ, θG]^JGHNPR\*NtOHQHQ9F=D&G=D&RTN^JG=D Y5OHNPYI&JIGHIR¦b5FHGHNjm*Nj\=b&n.JIXRTILM9OHIWI]PFHI\*l&stIW.DJl]rKθD&FHFHGHIY]PQHRaD&HQHN^dQzQH]PYDJlSQHOJNPGHNP]^bf[iQ=NcO=D&]VD&G=D&YJNPGHQH]^J.QoYHQyZbOHYHHQHQuG=D&]PFHFGHξ,Njm*θ Nj(x)\*NPOH≈QK G(x) WJNPGHRTQHO=DVyZbOHYHHQHQuG=D&]PFHGHNjm*Nj\*NPOHQK *çm*GbUPId]^JIGHIOH_n=WJNPIGHN^JQ=kHNPF]tξ,YIθ d]^JDJQH]PJQ=YNX]PkHQJ&D&N^J]Kn*kJIzIHNPOHYD Gn (X)OHNPQstWNP]^J.OHIUPI9F=D&G=D&RTN^JG=D KHW\=KHN^J]K'V.IGHI&pNPdn=NP]j\*QuFHGHQQHRTNPTNjJ9RTNjoθn→∞]^JIy]eVItm*QHRTI]^Jl¦D&YHQHNSIHNPOHYHQqO=Dst_-W.D&f[J]rKPTn (X) −→ θQqO=DV.ILm=KJ'pQHGHIYIN5FHGHQHRTNPOHNPOHQHN5WyRaDJNPRaDJQHkHNP]PYId ]^JDJQH]^J.QHYN&çbiNP]^J.W&b.oN^J+RTOHILMNP]^JWIQHOH_:VFHGHQHRTNPGHIWFHGHQHRTNPOHNPOHQK`G=Ds^\*QHkHOH_:VWQm*IWu]eVItm*QHRTI]^JNPd]j\=bk=D&dHOH_:V'WNj\*QHkHQHO'YD&Y"WSJNPIGHQHQnJD&Y'Q"O=DXFHG=D&YJQHYN&HCEDpNPd"sLDmDLkHNPd"WO=Do]^JIKiNPNWGHNPR¦K+OHNSKHW\=KHN^J]rK)bUe\=bv\*NPOHOHIN5Qs^bkHNPOHQHN5WI&stRTILM9OH_:V+FHI]^JD&OHIWIYsLDmDkO=D]eV.ILm*QHRTI]^JlH¾D&RTN^JQHR)\*QplHnkJIvI\*l&pIN-YI\*QHkHNP]^JWIEIFHGHNjm*Nj\*NPOHQHd]rV.Itm*QHRTI]^JNPd]\=bk=D&dHOH_:VWNj\*QHkHQHOXKHW\=KHN^J]rKSIFHG=D&WmD&OHOH_-RQXYHG=D&dHOHNW.DM9OH_-R-n-^!BIE!$& 7'¾ x%ÄSxTxË TIYDstDJlHnakJI Qs]eV.Itm*QHRTI]^JQ]j\*Njm=bN^J]eV.ILm*Qo(r)ξn −→ ξRTI]^Jl jPξn −→ ξ n → ∞ÁaCE -ÄE\=K"m*IYDsLDJNj\*l]^JW.D5bJWN^GM5m*NPOHQKysLDmDLkHQ'm*I]jJ&DJIkHOHIzWI]joFHI\*l&stIW.DJl]rK'&&n]PFHG=D&WNjmH\*QHW_-RûmH\=KW]PNjVQhom Û / / ?!!$& B & / $0k-+ε>0r>0[P(|ξn − ξ| ≥ ε) ≤E|ξn − ξ|r.εr¾ x%ÄSxTx» HIYDsLDJlHnHkJIQsE]eVItm*QHRTI]^JQOHN]j\*Njm=bN^J9]eV.ILm*QoPξn −→ ξRTI]^Jl j(r)ξn −→ ξ n → ∞ÁaCE -PÄSI]^JDJIkHOHI[FHGHQHWNP]^JQSFHGHQHRTNPGXFHI]j\*Njm*IWDJNj\*lOHI]^J.QEm*QH]PYHGHN^JoOH_:V ]j\=bk=D&dHOH_:VWNj\*QHkHQHOnk=D&]^JOH_-N5stO=DkHNPOHQK YI&JIGH_:VWI&stG=D&]^JD&f[Jv_-]^JGHNPNom s jjt ©:®¬{²¿Ù Î Úe¬®v¤:ÚÙ9z«jàqÙ w Æß.ÞàHv² Î ª« w ®ß&®H P] II&JWN^J]^J.W&bf[iQVk=D&]^JOH_:VstO=DLkHNPOHQHd`WNPGHILKJOHI]^JQnOHIu]tD&RTQ]j\=bk=D&dHOH_-N9WNjo\*QHkHQHOH_]eV.ILm=KJ]rKFHIcWNPGHILKJOHI]^JQ&Á¦D&]P]PRTI&JGHQHR)WNPGHILKJOHI]^JOHINFHGHI]^JG=D&OH]^JWInHUrm*N {}RaN^G=DÃXNPvNPUjDb]^Jl([0, 1], B[0, 1] , λ)λξn =¤kHNPWQm*OHIHn=kJIFHGHQPξn −→ 0en ,0,N^]j\*QNP]j\*Q 0 ≤ ω ≤ 1/n;vI\*NPNJI&UPIHnE|ξn |r =1/n < ω ≤ 1.ξn −→îï ðï 0 jnHOHIern−→ ∞nHm \=K'\*fcvIUPIr>0¾x%ÄSxax» @ IYDsLDJlHnkJI5QsE]eV.ILm*QHRTI]^JQOHN]j\*Njm=b.N^Jz]rV.Itm*QoPξn −→ ξRTI]^Jl jξn −→îï ðï ξ n → ∞ÁaCE -b]^Jl[v.DstIWIN¦WNPGHILKJOHI]^JOHINFHGHI]^JG=D&OoAA]^JWIQHRTNPN^JzWQmh {äIYHGbM9OHI]^Jl5Njm*QHOHQHkHOHIdA(Ω, F, P)mH\*QHOH_n {¯vIGHNj\*NPW]PYDK oÓΩD\*UPNPvG=D-FHILm*RTOHILMNP]^JW n {FσRTNPG=DÃXNPvNPUjDzO=D5|j\*NPRTNPOJDV¾D&ZEQHY]PQHGb.NPR¸OHNPYI&ΩJIGPbfJIkHYbO=DIYHGbM9OHI]^JQuQyFFHGHIQstWI\*lOHINEO=D&FHG=D&W\*NPOHQHN&ωA0A¤cJt\*ILM9QHRÀI&JJIkHYHQWsLDmD&OHOHIR;O=D&FHG=D&W\*NPOHQHQ¨m=bU^bA0mH\*QHOH_nHIvI&stO=DLkHQHR¨W&JIGHIduYIOHNP'm=bUPQukHNPGHN^snsLDoJNPR I&JJIkHYHQI&Jt\*ILM9QHRqW1/2sLDmD&OHOHIRqO=D&FHG=D&W\*NPOHQHQm=bU^bXmH\*QHOH_n&W&AJ1IGHIdYIOHNP m=bUPQIvAI1stO=DkHQHR1kHNPGHN^snI&JJIkHYHQI&Jt\*ILM9QHR¯m=bU^b 1/3 mH\*QHOH_Q'J&D&Y'mD\*NPN&ÄD&OHOH_-duFHGHIHANP2]P]XQstIvG=DMNPAOu2O=DGHQH]^bOHYN9 A2 A31/4I]^JGHIQHR¨]j\=bk=D&dHObf¥WNj\*QHkHQHOb :FHI5FHG=D&WQ\=bξn n = 1, 2, .
. .NP]j\*Qm=bUjDω∈1,NP]j\*Qm=bUjD An−1 An ;ξn =n→∞3om 02Ïd»¼±º ££¾ ¤01[ω 6∈0,An−1 An .I]PYI\*lYbymH\*QHOH_1m=bU]^JGHNPR¦KJ]KqY+Obt\*fSnQHRTNPNPRFHGHQ¾DoPZEQHY]PQHGbNPRÀFHGHIQstWI\*lObfûJIkHYbI]PYI\*ξnlY−→b§U^D&0GHRTIOHQHnkHNP→]PYHQH∞d¸GKmω0 ∈ ΩG=D&]eV.ILm*QJ]rKn.mD&OHObfJI&kHYbzFHIYHGHIN^JSvNP]PYIOHNPkHOHIN[RTOHIMNP]^JWIm=bU.¦DoAi−1 AiYHQHR¯IvG=DstIR-nW"mD&OHOHIdqJIkHYNzvNP]PYIOHNPkHOHINzRTOHILM5NP]^JWIy|j\*NPRTNPOJ.IWFHI]j\*Njm*IoW.DJNj\*lOHI]^JQv&bLm*N^JqG=D&WOKJl]rKYD&YObt\*fSnJD&Y§QNjm*QHOHQHHN&nξ1 (ω0 ), ξ2 (ω0 ), . .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
