А.А. Кудрявцев - Пособие по теории вероятностей (1115312), страница 13
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. . , ξ nWI'G=D&]PFHGHNjm*Nj\*NPOH_EnFHGHQHkHNPRnP(ξ1 = 1) = P(ξ1 = −1) = 1/4 P(ξ1 = 0) = 1/2CED&dJQ)RaDJNPRaDJQHkHNP]PYINILM9QmD&OHQHNXQum*QH]PFHNPGH]PQHf;]j\=bk=D&dHOHId)WNj\*QHkHQHOH_Sn =ξ1 + . . . + ξ nÁaCE -CED&dm*NPR¨RTIRTNPOJ_1]j\=bk=D&dHOHIdWNj\*QHkHQHOH_=RTNPNPRom ξ1Eξ1 = 1 ·[111+ 0 · − 1 · = 0,424Dξ1 = Eξ12 = 1 ·111+0· = .222I%]PWIdH]^JW&b RaDJN^RDJQHkHNt]PYIUPIcILM9QmD&OHQKnPFHI]PYI\*lYb]j\=bk=D&dHOH_-NWNj\*QHkHQHOH_Itm*QHO=D&YIWI5G=D&]PFHGHNjm*Nj\*NPO=_nξiESn = EnXξi =i=1nXi=1Eξi = n · 0 = 0.GHQ FHNPGHNjV.Itm*NzI&J'm*QH]PFHNPGH]PQHQ]^bRTRT_¥Y ^] bRTRTNm*QH]PFHNPGH]PQHdRT_¥WI]PFHI\*l&sPbNPRT]rKOHN^sLD&WQH]PQHRTI]^Jlfê]\=bk=D&dHOH_:VzWNj\*QHkHQHO I]PWIdH]^JW&b£cm*QH]PFHNPGH]PQHQzFHI\=bk=D&NPRξi[DSn = DnXξi =nXi=1i=1Dξi = n ·1n= .22¾ x%ÄSxTx Ö £H-ça\=bk=D&dHOH_-NWNj\*QHkHQHOH_OHN^sLD&WQH]PQHRT_æQêQHRTNPf[Jη1 , η 2 , .
. .ILm*QHO=D&YIWING=D&]PFHGHNjm*Nj\*NPO=QHN&nD]j\=bk=D&dHOH_-NSWNj\*QHkHQHOH_IFHGHNjm*Nj\=Koξif[J]K¨G=D&WNPOH]^Bi(1,JW.D&RTQ 1/2)nIYDsLDJlHnakJI]j\=bk=D&dHOH_-Nξi = ηi − ηi+2 i = 1, 2, . . .WNj\*QHkHQHOH_Itm*QHO=D&YIWIG=D&]PFHGHNjm*Nj\*NPOH_En:FHGHQHkHNPRQHRTNPN^J`JD&YINyM5Nξ1 , ξ 2 , . . .ξ1G=D&]PFHGHNjm*Nj\*NPO=QHN&n*YD&YyQyWsLDmDLkHN Ö &@ HIYDsLDJlHnHkJI5]j\=bk=D&dHOH_-NXWN\*QHkHQHOH_[ξis ê<t!ý Ú ¤:«¥z9²¿Ù w Æß.ÞàHv² Î ª« w ®ß&®H QOHN^sLD&WQH]PQHRT_;FHGHQnCED&dJQRaDJNPRaDJQHkHNP]tYIN"ILM9QmD&OHQHN5Qξi+jj ∈ IN j 6= 2*m QH]PFHNPGH]PQHf¿]j\=bk=D&dHOHIdWNj\*QHkHQHOH_nSn = ξ 1 + .
. . + ξ n n ≥ 2ÁaCE -CED&dm*NPRLI]PYI\*lYb]j\=bk=D&dHOH_-NWNj\*QHkHQHOH_OHN^sLDoP(ξ1 = 1)ηiWQH]PQHRT_n=QHRTNPNPRom 1P(ξ1 = 1) = P(η1 − η3 = 1) = P(η1 = 1, η3 = 0) = P(η1 = 1)P(η3 = 0) = .4xcO=D\*IUPQHkHOHIzFHI\=bk=D&NPRQP(ξ1 = 0) P(ξ1 = −1)ÄE\=KFHGHIW&NPGHYHQ§OHN^sLD&WQH]PQHRTI]^JQm*QH]PYHGHNjJ.OH_:V§]j\=bk=D&dHOH_:V§WNj\*QHkHQHOQζ1 ζ2m*I]^JDJIkHOHIFHIYDsLDJlHnkJImH\=K)W]PNjVQ JI]j\*Njm=bN^JzQP(ζsG=1D&]P=]^bxM51m*, NPζOH2QH=dn=xD&2O=)D\*=IUPP(ζQHkHOH1 _:=V'xFH1GH)P(ζIW&Njm*2 NP=OHOHx_-2RgFHGHQx1 x2GHN^pNPOHQHQusLDmDkHQ+ *k=D&]^JOHI]^JQn=mH\=KQHRTNPNPR!p[j 6= 2P(ξi = 1, ξi+j = −1) = P(ηi = 1, ηi+2 = 0, ηi+j = 0, ηi+j+2 = −1) == P(ηi = 1)P(ηi+2 = 0)P(ηi+j = 0)P(ηi+j+2 = −1) =1 1· =4 4= P(ξi = 1)P(ξi+j = −1).ÄE\=K9I]^JD\*lOH_:V9stO=DkHNPOHQHd']j\=bk=D&dHOH_:V"WNj\*QHkHQHOQ]PFHG=D&WNjmH\*QHW_êD&O=D\*Ioξi ξi+jUPQHkHOH_-NXG=D&WNPOH]^JW.D¿GHN^pNPOHQHQ`sLDmDLkHQ Ö &@ RT_;FHIYDsLD\*QnkJIÂ]PILM5D\*NPOHQHfSnRT_OHN"RTIMNPRWI]PFHI\*l&stIW.DJl]rK§]PW&IdH]^JWIR¿£m*QH]PFHNPESGH]PQHn Q=mH\=0KO=DV.IM5m*NPOHQKnDSnFHI]PYI\*lYbnIkHNPWQm*OHIHn]j\=bk=D&dHOH_-NWNj\*QHkHQHOH_QOHN%KHW\=KHf[J]rK"OHN^sLD&WQH]PQoξi ξi+2RT_-RTQg|^JI"\*NPUPYIyFHGHIWNPGHQJlHnFHIYDsLD&WHnO=D&FHGHQHRTNPGnkJIi = 1, ξi+2 = 1) 6= j*CED&dm*NPR=GHQu|^JIR¸v&bLP(ξm*NPRgbkHQJ_-W.DJ.lHn*kJIP(ξi = 1)P(ξi+2 = 1)ESn2 = DSn]j\=bk=D&dHOH_-NWNj\*QHkHQHOH_]^bJl 2 FHGHQQ9G=D&WOKHf[J]rK"FHISG=D&]PFHGHNjm*Nj\*NPO=QHfξi ξjξii=jFHGHQ=RTNPNPR2[η1 η3 − η 3 − η 1 η5 + η 3 η5|i − j| = 2ESn2 = EnXi=1= EXξi ξj +i, j: i=jξi!2X= EnXi=1ξi ξj +i, j: |i−j|=2ξi ·nXj=1ξj =Xi, j: |i−j|6=0, 2ξi ξj == nEξ12 + 2(n − 2)E(η1 η3 − η32 − η1 η5 + η3 η5 ) + (n2 − n − 2(n − 2))Eξ1 Eξ2 =11 1 1 1= n · + 2(n − 2)− − ++ (n2 − 3n + 4) · 0 = 1.24 2 4 4¾x%ÄSxax Ö ¢¦cb.]^Jlq]j\=bk=D&dHO=DKWNj\*QHkHQHO=DG=D&W&O=D+]^bRTRTN'IkHYIWHnFHIoηnKHWQHW&pQV.]rKFHGHQOHN^sLD&WQH]PQHRT_:VvGHI]tD&OHQKV¸FHG=D&WQ\*lOHIdQHUPG=D\*lOHIdYI]^JQn[b!cWdef0g7iFjj]PFHI\*l&sPb&KOHNPG=D&WNPOH]^JWI¨NPv_pNPW.D Ö . jn:IHNPOHQJl§]PWNPGVb¨mH\=KOHNPYI&JIGHIUPIε>0 ηnP − 3.5 ≥ ε .om nÁaCE -Tb]^Jl{ÑYI\*QHkHNP]^JWIqIkHYIWHnW_-F=D&W&pNPN"FHGHQ oIRvGHI]tDoξiOHQHQH:I]PFHI\*lstIW.D&W&pQH]PlY\D&]P]PQHkHNP]^Y=QHRgIFHGHNm*Nj\*NPOHQHNtR§WNPGHIKJOHI]^Ji QnFHI\=bk=DoNPR-n%kJIgFHGHQêILm*OHIRävGHI]tD&OHQHQêQHUPG=D\*lOHId¯YI]^JQ¯]PGHNjm*OHNPNkHQH]j\*IgW_-F=D&W&pQVG=D&WOKHN^J]rK*I]PYI\*lYbIkHYIW3.5EξiηnE =EnPni=1 ξin=3.5n= 3.5,nQuWWQm=b'OHN^sLD&WQH]PQHRTI]^JQ+]j\=bk=D&dHOH_:VuWNj\*QHkHQHOFHI\=bk=D&NPRηnD =DnPi=1 ξin=Pni=1 Dξin2= ηnP − 3.5 ≥ ε ≤nξinHQsXYI&JIGHI&d]j\*Njm=bN^Jn(91/6 − (21/6)2 ),n2105.36nε2M5~_7 77[¾x%ÄSxTx Ö *IYDsLDJlz]^FHG=D&WNjmH\*QHWI]^Jl9sLD&RTNPk=D&OHQK Ö ¾x%ÄSxTx Ö ÄSIYDsLDJlz]^WIdH]^JW.Du^o @ m*QH]PFHNPGH]PQHQ¾x%ÄSxTx Ö &Ö ÄSIYDsLDJlzGHNjsPbL\*lLJ&DJ_En*FHGHQHWNjm*NPOHOH_-NSWJ&D&v\*QHHN Ö ¾x%ÄSxTx Ö b]^Jl {û]j\=bk=D&dHO=DK)WNj\*QHkHQHO=D']YIOHNPkHOHId+m*QH]PFHNPGH]PQHNPdJ&D&YDKn=kJIÄSIYDξsLDJlHn=kJIDξ ≥ Eξ 22E|ξ| ≤ Dξ + 1¾x%ÄSxTx Ö @ ¦b]^Jl{ý]j\=bk=D&dHO=DK§WNj\*QHkHQHO=D+JD&YDKnkJIξP(0 < ξ <ÄSIYDsLDJlHnHkJI1) = 1Dξ < Eξ¾x%ÄSxTx Ö @ =b]^Jl {}FHIkJQO=D&WNPGHOHINEIUPG=D&OHQHkHNPOHO=DKu]j\=bk=D&dHO=DKWNjo\*QHkHQHO=DhÄSIξYDsLDJlHn=kJIP(|ξ| ≤ c) = 1Dξ ≤ cE|ξ|¾x%ÄSxTx Ö @ CED&dJQ)RaDJNPRaDJQHkHNP]PYINILM9QmD&OHQHNSQm*QH]PFHN^GH]PQ=f;]j\=bk=D&doncIFHGHNjm*Nj\*NPOHOHIdÇO=D¸WNPGHILKJOHI]^JOHIRûFHGHI]^JG=D&OH]^J.WNOHIdWNj\*QHkHQHOH_ξ = ξ(ω)n=Urm*Nnn{ÂRTNPG=D5ÃXNPvNPUjDn*NP]j\*Q+D. ~(Ω, F, P)ξ = ω2v ~=ΩW = [0, 1] F ~H=UL B[0, 1] P = λ ξ = ω − 1/2ξ = sin πωξ = sin 2πω¾x%ÄSxTx Ö @@ jÄSQ=D&RTN^JGXYHGbUjD QstRTNPGHNPOXFHGHQHv\*QMNPOHOHIHnLQQstWNP]^JOHI%\*QplHnkJI¦ç:kHQJ&DK]j\=bdk=D&dHOHIdWNj\*QHkHQHOHIdnG=D&WOHIRTNPGHOHI+G=D&]PFHGHNjo0 < a ≤ d ≤ bdm*Nj\*NPOHOHIdO=D'I&JGHN^stYNnO=D&dJQRaDJNPRaDJQHkHNP]PYINzILM9QmD&OHQHNQ)m*QH]PFHNPGH]PQHf[a, b]F\*I&iDm*QYHGbUjD[[[[s ê<t!ý Ú ¤:«¥z9²¿Ù w Æß.ÞàHv² Î ª« w ®ß&®H[¾ x%ÄSxax Ö @ £H=b]^Jl"]j\=bk=D&dHOH_NWNj\*QHkHQHOH_IFHQH]P_-WD&f[J"GHNjoξ1 , .
. . , ξn+1sPbt\*lJDJ_QH]PFH_JD&OHQHdz :NPGHObt\H\*Q5]:WN^GHIKJOHI]^Jlfgºb]PFHNjVD Q]j\=bk=D&dHO=DKpWNj\*QHkHQHO=D n+1G=D&WO=DkHQH]j\=b'J&D&YHQV jn=kJI=]PFHI\*loηni i = 1, . . . , nξi = ξi+1 = 1sPb&K¨OHNPG=D&WNPOH]^JWINPv&_pNPW.D Ö . jn¦IHNPOHQJl`]PWNPGVbmH\=KP(|ηn /n − p2 | ≥ ε)W]PNjVε>0¾x%ÄSxax Ö @ ¢b]^Jl Q {¼OHN^sLD&WQH]PQHRT_-N]j\=bk=D&dHOH_-N5WNj\*QHkHQHOH_Ç]YIoOHNPkHOH_-RTQym*QH]PFHNPGH]PQKHRTQHÄSIξ YDsLηDJlHn=kJIDξη ≥ Dξ · Dη¾x%ÄSxax Ö @ b]^Jl { ]j\=bk=D&dHO=DK)WNj\*QHkHQHO=D9]SYIOHNPkHOH_-RkHN^JWNPGJ_%RξRTIRTNPOJIR-ç:G=D&WOHQJl4 QEξ(Eξ)4¾x%ÄSxax Ö @ =b]^Jlz]j\=bk=D&dHOH_-NXWNj\*QHkHQHOH_Q OHN^sLD&WQH]PQHRT_¯Qnξ ηEξ = 1nQTCED&dJQ§RaDJNPRaDJQHkHNP]tYHQHNuILM9QmD&OHQK]j\=bk=D&dHOH_:VEη = 2 Dξ = 1 Dη = 4WNj\*QHkHQHOD. 2~Hv ξ + 2η 2 − ξη − 4ξ + η + 4(ξ + η + 1)2¾x%ÄSxax Ö @Ö aÄSIYDsLDJlHnkJImH\=K¨\*fcv_:V]j\=bk=D&dHOH_:VêWNj\*QHkHQHOQ nξηQHRTNPf[iQVuYIOHNPkHOH_NEm*QH]PFHNPGH]PQHQn*]PFHG=D&WNjmH\*QHW_1OHNPG=D&WNPOH]^JW.DqDξ −qDη2q≤ D(ξ + η) ≤Dξ +qDη2.¾x%ÄSxax Ö @ aÄSIYDsLDJlHnkJImH\=K¨\*fcv_:V]j\=bk=D&dHOH_:VêWNj\*QHkHQHOQ nξηQHRTNPf[iQVuYIOHNPkHOH_NSW&JIGH_-NERTIRTNPOJ_n]PFHG=D&W&NjmH\*QHWI9OHNPG=D&WNPOH]^JWIE|ξη| ≤qEξ 2·qEη 2 .¾ x%ÄSxax Ö £ =IYDsLDJlHnHkJImH\=K"\*fcvIdyHNj\*IkHQH]j\*NPOHOHId]j\=bk=D&dHOHIduWNjo\*QHkHQHOH_ nQHRTNPf[iNPdYIOHN^kHOH_-dW&JIGHId¸RTIRTNPOJLn:]PFHG=D&WNjmH\*QHWI`OHNPG=D&WNPOH]^JWIξEξ(1 − Eξ) ≤ Dξæä æ¿ÌÅ÷ Ûã äèËÊ`÷ Ñè Ïã¸ÊÌèã èÈäßèÅÎ B ßâáýöè ã ¹ ·Ñç Ìèèñûó ¡ø ùûúýüËþ¿ÿ b]^Jl"sLDmD&OHI9OHNPYI&JIGHINSWNPGHIKJOHI]^JOHINFHGHI]^JG=D&OH]^J.WIQ]j\=b.o(Ω, F, P)k=D&dHO=DKgWNj\*QHkHQHO=DO=DqOHNPR-aCED&G.Km=b]uWN^iN^]^J.WNPOHOH_%RTQ¸]j\=bk=D&dHOH_-RTQ¨WNj\*QokHQHO=D&RTQzJD&YMN[RTILM9ξOHIEG=D&]P]PRaDJGHQHW.DJlSYIRTF\*NPY]POHI&stO=DLkHOH_-Nc]j\=bk=D&dHOH_-NcWNj\*QokHQHOH_nFHIOHQHRaDKgFHItm|^JQHR;ZbOHYHHQHfnTUrm*N{ ]j\=bk=D&doξ1 (ω) + iξ2 (ω)(ξ1 , ξ2 )OH_-d¨WNPYJIGFHItm*GHIvOHNPNuI ]j\=bk=D&dHOH_:V¨WNPYJIG=DVgGHNPkHlFHIdm*N^JW jTGHQ|^JIRgFHG=D&WQ\DG=D&vI&J_]PI]j\=bk=D&dHOH_-RTQyWNj\*QHkHQHO=D&RTQyI]^JD&f[J]rKuOHNPQstRTNPOHOH_-RTQnD5RTOHQHRaDKyNjm*QHOHQH=D G=D&]P]PRaDJGHQHW.D&N^J.]rK+YD&YyIv_-kHO=DKuYIOH]^JD&OJ&Di¤EÁ Ä ¦Ã aC § T#E=B'"!'U>=*S10EH!E=!'[1\EH<>!1*S1B!'.`^=C5D 7*AA)+=\U^-\;+ 7!*^O><!)*U^n^O0-HE!% =E!'<^n)+#-1*"+'!*k\vIE' P#[AB. ><'!!!$ ξP# vϕ;+$5: +IR#−→Ct ∈ IRξϕξ (t) = Eeitξ=Zeitx dFξ (x).S VA- / ;+!'\ åH-I.#|h"'"I- SA- / ;+!'GåH-If (x)(9.1)¾x axCE ; :¡SbOHYHHQKnaIFHGHNjm*Nj\=KHNPRaDK¸]PII&JOHI&pNPOHQHNtRæ ϕξ (t)=O Dst_-W.D&N^J]rK"JD&YMN-ZbOHYHHQHQ T]joFξ (x)\*Q"ZbOHYHHQKQHRTNPN^JF\*I&JOHI]^Jln.JIEVD&G=D&YJNPGHQH]PJQ=kHNP]PYDK"ZbOHYHHQKFξ (x)NP]^JlZbOHYHHQHQ fξ (x) hξS [ ïϕξ (t) =Zeitx fξ (x) dx.¾x axCE  D&G=D&YJNPGHQH]^J.QHkHNP]tYDK1ZbOHYHHQK¯W]PNPUemDg]^biNP]^J.W&bNPJnFHI]PYI\*l&YbZZ itx|ϕξ (t)| = e dFξ (x) ≤ eitx dFξ (x) = 1. @s ð t Þ x ÞÛ¥z« x ®Ù-zL®ß«PÙÛH®«® Øx Ú®=Ý^ªÚe¬¥yS®«VÇÆHÛH®H®¾xSaxCE¿ @ ¦s']PWIdH]^JW)YIRTF\*NPY]POH_:V§kHQH]PNj\¨QIFHGHNjm*Nj\*NPOHQKg ]j\*Njm=bN^Jn*kJIVD&G=D&YJNPGHQH]^J.QHkHNP]tYDKZbOHYHHQKuJD&YMNEFHGHNjm*]^JD&WQHRaDzWzWQm*N&£ïϕξ (t) = E cos(tξ) + iE sin(tξ).(9.2)D&G=D&YJNPGHQH]^J.QHkHNt]PYDKZbOHYHHQKuIv\DmD&N^Jz]j\*Njm=bf[iQHRTQI]POHIWOH_-RTQu]PWIdo]^JW.D&RTQh 9mH\=K"\*fcvIdu]j\=bk=D&dHOHIdWNj\*QHkHQHOH_Qϕξ (0) = 1[ 9mH\=K"\*fcvIdu]j\=bk=D&dHOHIdWNj\*QHkHQHOH_ξ|ϕξ (t)| ≤ 1Q"\*fcv_:VξmH\=KuW]PNjVt;a, b ∈ IRϕaξ+b (t) = eitb ϕξ (at);@ 'NP]j\*QξQη]^bJl9OHN^sLD&WQH]PQHRT_-NX]j\=bk=D&dHOH_-NSWNj\*QHkHQHOH_n=JIϕξ+η (t) = ϕξ (t)ϕη (t);£ 9VD&G=D&YJNPGHQH]^J.QHkHNt]PYDKZbOHYHHQKKHW\=KHN^J]rKG=D&WOHIRTNPGHOHIuOHNPFHGHNPGH_Woϕξ (t)OHId~¢ 'NP]j\*QnnJI']^biNP]^J.W&bNPJuOHNPFHGHNPGH_-WO=DK oºK+FHGHIQstWItm*O=DKE|ξ|k < ∞ k ≥ 1kZbOHYHHQHQ=n FHGHQHkHNPRϕξ (t)(k)ϕξ (t)t=0= ik Eξ k ;. 9mH\=KYIRTF\*NPY]POHIE]PIFHGKMNPOHOHIdSVD&G=D&YJNPGHQH]PJQHk=NP]PYIdzZbOHYHHQHQ5]PFHG=D&WNjmHo\*QHWI5G=D&WNPOH]^JWIϕξ (t) = ϕξ (−t) = ϕ−ξ (t).@#E=B'"!'U>E=*O^0EH!E=.!!'<'^Ü9R BHE><1!*1ÜB'!><'!!$3*;+'< 7M*)-;+!><Q*A)+\U^ -A)+\.)+\ 1¤!E+Á'!P# ÄB¦ IEÃ!a$&%LCE><' -\Cl *[ aA,a)+B*, 'U.
!;. .!**!)-^O^!;+?)$<+^ 1Û0HE!^=E0!'BHE!13=E)!HEA'<6^?"\='< r)+ 7*A+ / =B;DIE VP# ["A' . ¤Á ux¥ " JNPIGHNPRaDzNjm*QHOH]^J.W&NtOHOHI]^J.Q* j01ψ : C −→ C|z| ≤ 1ψ(z) =∞Xn=0an z n .!b cWdef0g7iFj ¢.¢¤EÁĦÃaC @ ¥*'U;*)-^!?1O0HE!=E!'1M\*><'B!*1BHE><1!.*-*1(\'!'!!><'!!\$IEξ!C{$A'!;+!!'<>U 7!6'<?^UCY!1(;+$A+`98F'!^[0\HE!IE=E!!$&'< 7^~'(=DB 7A^!"=!VP#^< 7AB'\ '!k"!I .P# ψ : C −→ C Cl*A)+B!!*^Q)U^ |z| ≤ 1 \)HE6?"'< r / ; Vξψξ (z) = Eξ z .S¾x axCE ¯ £HÄE\=Kq]j\=bk=D&dHOHId WNj\*QHkHQHOH_nbLm*IW\*N^JWIG.KHf[iNPdb]j\*IoWQKHRIFHGHNjm*Nj\*NPOHQK @ Q+FHGHQHOHQHRaD&f[iNPd)stO=DLkHNPOHQK]WNPGHILKJOHI]^JLKHRTQ0, 1, .
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