概率论基础第5章

55.15.25.35.15.1.1w1Fn(x)→F(x)Fn(x)F(x)nFnFxFxxClim()(),→∞=∈5.1.1Fn(x)=0,xn1,xnxFn(x)0F(x)≡0Fn(x)FFF(x)FCFFCFCF5.1.2ξn=-⎯ξ=0ωξn(ω)→ξ(ω)0,x-⎯1,x-⎯F(x)=1n1n0,x01,x0Fn(x)=1nFn(0)=1F(0)=02Lévy-CramerFn(x)F(x)Fn(x)fn(t)f(t)fn(t)F(x)f(t)Fn(x)F(x)Fn(x)fn(t)F(x)f(t)tfn(t)f(t)f(t)t=0fn(t)Fn(x)F(x)f(t)F(x)5.1.21Fn(x)→F(x)wξn→ξdξn→ξpnnlimP{||}0ξξε→∞−≥=rξn→ξrrnnElim||0ξξ→∞−=r0ξn→ξa.snnP{lim()()}1ξωξω→∞==2.5.1.1Remarkξn→C⇔ξn→Cpd5.1.3Cfn(t)eiCt5.1.4ξn→ξξn→ηP(ξ=η)=1pp3.5.1.2r.MarkovIneq∀n∈Nξn(ω)=n1/r,ω∈[01/n)0,ξn→0Eξnr≡n×1/n=1pξn[01)5.1.31{An}AnnnnknkAA1lim∞∞→∞===∩∪{An}AnnnknknAA1lim∞∞==→∞=∪∩2Borel-Cantelli(1){An}nnAP{lim}0→∞=nnA=1P()∞+∞∑(2){An}nnA=1P()∞=+∞∑nnAP{lim}1→∞=1.(1)nnnnkknknkAAA1P{lim}P{}limP{}∞∞∞→∞→∞=====∵∩∪∪nknkAlimP()0∞→∞=≤=∑(2)1-xe-x0x1nnkAnnnknkAAeP()0P{}[1P()]0∞=∞−∞==∑≤=−≤=∏∩nnnknkAieA1P{}0;.P{lim}1∞∞→∞==∴==∪∩nnnmknkm111{:lim()()}{:(|()()|)}ωξωξωωξωξω∞∞∞→∞=====−∩∪∩ωmNnN|ξn(ω)-ξ(ω)|1/m()nmknkm111P{(|()()|)}0ξωξω∞∞∞===−≥=∪∩∪3ε0nknk1P{(|()()|)}0ξωξωε∞∞==−≥=∩∪nknk1P{(|()()|)}1ξωξωε∞∞==−=∪∩ε0NnN|ξn(ω)-ξn(ω)|εω1ε0nknklimP{(|()()|)}0ξωξωε∞→∞=−≥=∪5.1.3ξn→ξa.snn1P{|()()|}ξωξωε∞=−≥+∞∑5.1.4knnk(|()()|){(|()()|)}ξωξωεξωξωε∞=−≥⊂−≥∵∪5.1.41.ξn⎯→ξg(x)g(ξn)⎯→g(ξ)p(d)p(d)2.ξn→ξηn→ηξn±(×)ηn→ξ±(×)ηppp3.ξn→ξE(⎯⎯⎯⎯⎯)→0p|ξn-ξ|1+|ξn-ξ|4.ξn→ξξnk→ξpa.s5.ξn→0ξn↓0ξn→0pa.s5.11-2.357133-4.5.1.45.2(LLN)5.2.1{ξn}{ξn}npkkkEn11()0ξξ=−→∑5.2.2BernoulliµnnBernoulliAP(A)=p⎯→pµnnp.ChebyshevIneqnnpDnnn2211P{||}()4µµεεε−≥≤≤Chebyshev{ξn}DξnCnpkkkEn11()0ξξ=−→∑Markov{ξn}MarkovnkkDn211()ξ=→∑npkkkEn11()0ξξ=−→∑05.2.3Khintchine{ξn}Eξn=µnpkkn11()0ξµ=−→∑.{ξn}µµ5.1.2eiµtξnf(t)=1+iµt+o(t)nkkn11ξ=∑nnittttfioennn[()][1()]µµ=++→Remark“”KolmogrovKhintchine5.2.3Kolmogrov{ξn}naskkkEn.11()0ξξ=−→∑nnDn21ξ∞=+∞∑Kolmogrov{ξn}∀ε0kniikknikED21111P{max|()|}ξξεξε≤≤==−≥≤∑∑KolmogrovEξk=0Sk=ξ1+ξ2+…+ξknAk={ω:|Sk|ε|Sj|ε1jk-1}YkAkΣkn=1Yk1nnkkkknkkSSA111{max||}{||}εε≤≤==≥=≥=∑∪Ak|Sk|εnnnkkkkkknkkkESYSAEY221111()P{max||}P()εε≤≤===≥==≤∑∑∑Sn-SkSk×YkE(Sk2×Yk)E(Sk2×Yk)+E[(Sn-Sk)2×Yk]+2E[(Sn-Sk)×SkYk]=E(Sn2×Yk)nknknnknkSESYESDS2222211111P{max||}()εεεε≤≤=∴≥≤≤=∑5.21-4.35752240465.3(CLT){ξn}Eξk=µkDξk=σk2Sn=ξ1+…+ξnBn2=DSnnkknnnknnkknnSESBDS1,ξµξηξ=−−===∑ηnN(01){ξn}nnnSESxxxDS1P{}(),RΦ∀−→∈5.3.1CLT1Lévy-Lindeberg{ξn}µσ2{ξn}CLT.ξk-µf(t)ηnntfn[()]σξktttfonnn22()1()2σ=−+2DeMoivre-LaplaceµnnBernoulliAP(A)=pnnpxxxnpq1P{}(),RµΦ∀−→∈Remark()kknpxnpq−=kxnkenpq21211P{}/()12µπ−=→5.3.2CLT1LindebergξkFk(x)∀τ0knnkkxBnknxdFxB22||11lim()()0µτµ−≥→∞=−=∑∫Lindebergnkkn1P{max||}0ξτ≤≤≥→ηn“”LindebergAk={|ξk-µk|τBn}1knnkkknknknB11P{max||}P{max||}ξτξµτ≤≤≤≤≥=−≥nnnkknkkkkkBAA111P{||}P{}P()ξµτ====−≥=≤∑∪∪knnkkxBdFx1||1()µτ=−≥=∑∫knnkkknxBxdFxB2221||11()()µτµτ=−≥≤×−∑∫Lindeberg{ξn}Lindeberg{ξn}CLT.Eξnk=0Dξnk=σk2/Bn2FnkfnkξnkLindebergnnkxnkxdFx2||1lim()0;τ≥→∞==∑∫∀t∈R1tnnnkkftfte221()()−==→∏1.t∈R1x∈R1itxeitxtxtxtx22231|(1)|min{||,||}2−+−≤2.1z1…znω1…ωnnnnkkkkkkkzz111||||ωω===−≤−∏∏∑1FnkStieltjesknknknkntftEtEtB22232|()(1)|min{||,||}2σξξ−−≤τ0nknkxxtxdFxtxdFx23||||()()||()ττ≥+∫∫knknxtxdFxtB22232||()||,τστ≥≤+∫knknkkntftB2221|()(1)|2σ=−−∑nnnkkxtxdFxt,02231||()||0,τττ→∞↓=≥≤+→∑∫2t∈R1nnknkkkntftB22211|()(1)|0.2σ==−−→∏∏τ0knknknxDxdFxB2222||()τσξτ≥=≤+∫nknkknknxxdFxB222211||max()τστ≤≤=≥≤+∑∫LindebergknknnB221lim{m)}(ax0σ→∞≤≤∗→0x1/2|e-x-1+x|=|x2/2-x3/3!+x4/4!-..|x2/2(*)x=t2σk2/2Bn2kntnnBkkknteB222222211|(1)|2σσ−==−−∏∏kntnnBkkkknntteBB2222224222211|1|()028σσσ−==≤−+≤→∑∑kntnnBnkkkftet2222111|()|0,Rσ∀−==−→∈∏∏2CLT∃Knmax|ξk|Kn⎯→01knKnBn.∀τ0NnN2KnτBnkn{|ξk-µk|τBn}=ΩnknnkkknxBxdFxB221||1()()µτµ=−−∑∫nkkknxdFxB2211()()1µ+∞=−∞=−=∑∫LyapurovnkkknEB2211||0δδξµ++=−→∑∃δ0.LindebergknnkkknxBxdFxB221||1()()µτµ=−≥−∑∫knnkkknnxBxdFxBB221||11||()()δδµτµτ+=−≥≤−∑∫nkkknxdFxB22111||()0δδδµτ+∞++=−∞≤−→∑∫5.3.1ξk~U(-kk){ξn}CLT.ξk|ξk|nk=12…nσk2=k2/3Bn2=O(n3){ξn}CLT5.3.2(1)ξk~U(-k1/2k1/2)CLT(2)Lyapurov3Lindebergi.i.dLindebergCLTCLTσ2knnkkknxBxdFxB221||1()()µτµ=−≥−∑∫xnxdFx22||1()()0µτσµσ−≥=−→∫LindebergCLTLindebergLindeberg5.3.3ξk~N(µk2-k)ηn~N(01){ξn}CLTBn2=1-2-n1τ0knnkkknxBxdFxB221||1()()µτµ=−≥−∑∫xxdFxconstant1211||()()−≥≥−=∫µτµLindebergξkσk2“”knknnB212121max122σ−−≤≤=→−4FellerknknnB1limmax0σ→∞≤≤→nnnBB,0σ⇔→+∞→Feller{ξn}Lindeberg⇔{ξn}CLT+Feller5.3.3LLNCLT1.2.i.i.diffi.i.diff3.CLTLLNDSn=o(n2)CLT∀ε0nkkknnEnDS11P{|()|}2()1εξξεΦ=−≈−∑5.31-4.35814195160