glivenko-cantelli格里文科定理证明

1TheGlivenko-CantelliTheoremLetXi,i=1,...,nbeani.i.d.sequenceofrandomvariableswithdistribu-tionfunctionFonR.TheempiricaldistributionfunctionisthefunctionofxdefinedbyˆFn(x)=1nX1≤i≤nI{Xi≤x}.Foragivenx∈R,wecanapplythestronglawoflargenumberstothesequenceI{Xi≤x},i=1,...ntoassertthatˆFn(x)→F(x)a.s(inordertoapplythestronglawoflargenumbersweonlyneedtoshowthatE[|I{Xi≤x}|]∞,whichinthiscaseistrivialbecause|I{Xi≤x}|≤1).Inthissense,ˆFn(x)isareasonableestimateofF(x)foragivenx∈R.ButisˆFn(x)areasonableestimateoftheF(x)whenbothareviewedasfunctionsofx?TheGlivenko-CantelliThoeremprovidesananswertothisquestion.Itassertsthefollowing:Theorem1.1LetXi,i=1,...,nbeani.i.d.sequenceofrandomvariableswithdistributionfunctionFonR.Then,supx∈R|ˆFn(x)−F(x)|→0a.s.(1)Thisresultisperhapstheoldestandmostwellknownresultintheverylargefieldofempiricalprocesstheory,whichisatthecenterofmuchofmoderneconometrics.Thestatistic(1)isanexampleofaKolmogorov-Smirnovstatistic.Wewillbreaktheproofupintoseveralsteps.Lemma1.1LetFbea(nonrandom)distributionfunctiononR.Foreach0thereexistsafinitepartitionofthereallineoftheform−∞=t0t1···tk=∞suchthatfor0≤j≤k−1F(t−j+1)−F(tj)≤.1Proof:Let0begiven.Lett0=−∞andforj≥0definetj+1=sup{z:F(z)≤F(tj)+}.NotethatF(tj+1)≥F(tj)+.Toseethis,supposethatF(tj+1)F(tj)+.Then,byrightcontinuityofFtherewouldexistδ0sothatF(tj+1+δ)F(tj)+,whichwouldcontradictthedefinitionoftj+1.Thus,betweentjandtj+1,Fjumpsbyatleast.Sincethiscanhappenatmostafinitenumberoftimes,thepartitionisofthedesiredform,thatis−∞=t0t1···tk=∞withk∞.Moreover,F(t−j+1)≤F(tj)+.Toseethis,notethatbydefinitionoftj+1wehaveF(tj+1−δ)≤F(tj)+forallδ0.ThedesiredresultthusfollowsfromthedefinitionofF(t−j+1).Lemma1.2SupposeFnandFare(nonrandom)distributionfunctionsonRsuchthatFn(x)→F(x)andFn(x−)→F(x−)forallx∈R.Thensupx∈R|Fn(x)−F(x)|→0.Proof:Let0begiven.WemustshowthatthereexistsN=N()suchthatfornNandanyx∈R|Fn(x)−F(x)|.Let0begivenandconsiderapartitionofthereallineintofinitelymanypiecesoftheform−∞=t0t1···tk=∞suchthatfor0≤j≤k−1F(t−j+1)−F(tj)≤2.Theexistenceofsuchapartitionisensuredbythepreviouslemma.Foranyx∈R,thereexistsjsuchthattj≤xtj+1.Forsuchj,Fn(tj)≤Fn(x)≤Fn(t−j+1)F(tj)≤F(x)≤F(t−j+1),whichimpliesthatFn(tj)−F(t−j+1)≤Fn(x)−F(x)≤Fn(t−j+1)−F(tj).2Furthermore,Fn(tj)−F(tj)+F(tj)−F(t−j+1)≤Fn(x)−F(x)Fn(t−j+1)−F(t−j+1)+F(t−j+1)−F(tj)≥Fn(x)−F(x).Byconstructionofthepartition,wehavethatFn(tj)−F(tj)−2≤Fn(x)−F(x)Fn(t−j+1)−F(t−j+1)+2≥Fn(x)−F(x).Foreachj,letNj=Nj()besuchthatfornNjFn(tj)−F(tj)−2andletMj=Mj()besuchthatfornMjFn(t−j)−F(t−j)2.LetN=max1≤j≤kmax{Nj,Mj}.FornNandanyx∈R,wehavethat|Fn(x)−F(x)|.Thedesiredresultfollows.Lemma1.3SupposeFnandFare(nonrandom)distributionfunctionsonRsuchthatFn(x)→F(x)forallx∈Q.SupposefurtherthatFn(x)−Fn(x−)→F(x)−F(x−)foralljumppointsofF.Then,forallx∈RFn(x)→F(x)andFn(x−)→F(x−).Proof:Letx∈R.WefirstshowthatFn(x)→F(x).Lets,t∈Qsuchthatsxt.FirstsupposexisacontinuitypointofF.SinceFn(s)≤Fn(x)≤Fn(t)ands,t∈Q,itfollowsthatF(s)≤liminfn→∞Fn(x)≤limsupn→∞Fn(x)≤F(t).SincexisacontinuitypointofF,lims→x−F(s)=limt→x+F(t)=F(x),3fromwhichthedesiredresultfollows.NowsupposexisajumppointofF.NotethatFn(s)+Fn(x)−Fn(x−)≤Fn(x)≤Fn(t).Sinces,t∈QandxisajumppointofF,F(s)+F(x)−F(x−)≤liminfn→∞Fn(x)≤limsupn→∞Fn(x)≤F(t).Sincelims→x−F(s)=F(x−)limt→x+F(t)=F(x),thedesiredresultfollows.WenowshowthatFn(x−)→F(x−).FirstsupposexisacontinuitypointofF.SinceFn(x−)≤Fn(x),limsupn→Fn(x−)≤limsupn→Fn(x)=F(x)=F(x−).Foranys∈Qsuchthatsx,wehaveFn(s)≤Fn(x−),whichimpliesthatF(s)≤liminfn→∞Fn(x−).Sincelims→x−F(s)=F(x−),thedesiredresultfollows.NowsupposexisajumppointofF.Byas-sumption,Fn(x)−Fn(x−)→F(x)−F(x−),and,bytheaboveargument,Fn(x)→F(x).Thedesiredresultfollows.ProofofTheorem1.1:IfwecanshowthatthereexistsasetNsuchthatPr{N}=0andforallω6∈N(i)ˆFn(x,ω)→F(x)forallx∈Qand(ii)ˆFn(x,ω)−Fn(x−,ω)→F(x)−F(x−)foralljumppointsofF,thentheresultwillfollowfromanapplicationofLemmas1.2and1.3.Foreachx∈Q,letNxbeasetsuchthatPr{Nx}=0andforallω6∈Nx,ˆFn(x,ω)→F(x).LetN1=Sx∈Q.Then,forallω6∈N1,ˆFn(x,ω)→F(x)byconstruction.Moreover,sinceQiscountable,Pr{N1}=0.4Forintegeri≥1,letJidenotethesetofjumppointsofFofsizeatleast1/i.Notethatforeachi,Jiisfinite.NextnotethatthesetofalljumppointsofFcanbewrittenasJ=S1≤i∞Ji.Foreachx∈J,letMxdenoteasetsuchthatPr{Mx}=0andforallω6∈Mx,ˆFn(x,ω)−Fn(x−,ω)→F(x)−F(x−).LetN2=Sx∈JMx.SinceJiscountable,Pr{N2}=0.Tocompletetheproof,letN=N1∪N2.Byconstruction,forω6∈N,(i)and(ii)hold.Moreover,Pr{N}=0.Thedesiredresultfollows.2TheSampleMedianWenowgiveabriefapplicationoftheGlivenko-CantelliTheorem.LetXi,i=1,...,nbeani.i.d.sequenceofrandomvariableswithdistributionF.SupposeoneisinterestedinthemedianofF.Concretely,wewilldefineMed(F)=inf{x:F(x)≥12}.AnaturalestimatorofMed(F)isthesampleanalog,Med(ˆFn).UnderwhatconditionsisMed(ˆFn)areasonableestimateofMed(F)?Letm=Med(F)andsupposethatFiswellbehavedatminthesensethatF(t)12whenevertm.Underthiscondition,wecanshowusingtheGlivenko-CantelliTheoremthatMed(ˆFn)→Med(F)a.s.Wewillnowprovethisresult.SupposeFnisa(nonrandom)sequenceofdistributionfunctionssuchthatsupx∈R|Fn(x)−F(x)|→0.Let0begi