随机变量列的收敛性及其相互关系_郝锐利

:2005210212:167224291(2006)Sup.20010203(,710100):,,.:;;;:O21111:ASeveralconvergencesofsequencesofrandomvariablesandtheirrelatoinsHAORui2li(ChanganNormalSchool,Xian710100,Shaanxi,China)Abstract:Definitionsofconvergenceinprobability,convergenceindistributionandalmostsureconvergenceofsequencesofrandomvariablesaregiven.Byusingrelatedexamples,theseconceptsareexplained.Somerelationsamongthemarediscussed.Keywords:randomvariable;convergenceinprobability;convergenceindistribution;almostsureconvergence,,,.,(,,P),,..{n(n1)}(,,P).1()limnP{|n()-()|}=00,{n()}(ConvergenceinProbability)(),nP.(1),(1):n,n()().,nP0,N,nN|n-|.P{|n-|}0{|n-|}1.1[0,1],=[0,1),[0,1)Borel,PLebesgue,11=1,21=1,0,12,0,12,1.,k,[0,1)k,ki()=1,i-1k,1k,i=1,2,,k,0,.1=11,2=21,3=22,4=31,{n}.P0,P{|ki()|}=Pi-1k,ik=1k0,nP0.2()F(x)Fn(x)(n1),limnFn(x)=F(x),xCF.CFF,{Fn}F,FnWF.(2)34()Vol.34Sup.20066JournalofShaanxiNormalUniversity(NaturalScienceEdition)Jun.2006,(WeakConvergence).,,.,,{Fn}F.,FnWF,CFF,,.,F(x),,.3()nFn(x)F(x),Fn(x)WF(x),{n}(ConvergenceinDistribution),nL.2n,Fn(x)=0,x-1n,1,x-1n;F(x)=0,x0,1,x0.FnWF,FnFn=-1n=0,{n}.4()P{limnn()=()}=1,{n()}(AlmostSureConvergence){()},{n()}1(ConvergenceinProbability1)n(),n()a.s.().(3),{n()}P{limnn()=0}=1,(4){n()}0,n()a.s.0,{n()}()n-a.s.0.(),A0={|limnn()=0|},(4)P(A0)=1.(3),,{n()}n(),,P0,k1,nk|n()-()|,{n()}E=m=1k=1n=k:|n()-()|1m,n()a.s.(),P(E)=Pm=1k=1n=k|n-|1m=1,(5)P(ŠE)=Pm=1k=1n=k|n-|1m=0.(6)(5)(6):0,Pk=1n=k|n-|=0.(7)0,m=1k=1n=k|n-|1m=k=1n=k(|n-|).(6)(7).,0Pm=1k=1n=k|n-|1mm=1Pk=1n=k|n-|1m=0,(7)(6).(7)P0,Pk=1n=k|n-|=1.(8)Ak=n=k|n-|A1A2Ak.limkAk=k=1n=k|n-|.,(8)P0,limkP=n=k|n-|=1.DeMorgan,limkP=n=k|n-|=0.(9){n}a.s..3{n=n},P{n=n}=P{n=-n}=122n,Pn=1n=Pn=-1n=121-12n.0n1,Pm=n|n|m=n12mn0na.s.0.,{n}1:11,,P0,N0,mN,nN,|n-m|1,(Cauchy):P{limm,n(|n-|)=0}=1,{n},..:1nP,nL.Fn(x)F(x)n,yx,{y}={y,nx}{y,nx}{nx}{|n-|x-y}F(y)Fn(x)+P{|n-|x-y},n,nP0,F(y)limnFn(x)xzlimnF(z).xCF,yxzx,F(x)limnFn(x)limnFn(x)F(x),F(x)=limnFn(x),PxCF.nL...4P=12,n1n=,n{n},P{|n-|=1}=P{|-|=1}=12.{n},c,.2nPcZnLc.3,nPcnPc.P{|n-c|}=P{nc+}-P{nc-}=Fn(c+)-Fn(c-+0),nLc,limnFn(x)=1,xc,0,xc.limnP{|n-c|}=1,nPc.:3na.s.,nP.na.s.0,nP0.{|k|}={n=k|n|},1limkP{|k|}limkP{n=k|n|}=1.limkP{|k|}=1na.s.0nP0.3,,.:(,,P),,,P{}0,a.s..,:]].,.,nLc,c,nPc;nP,{nk}nka.s.;{n}nP0,na.s.0.:[1].[M].:,1999.[2].[M].:,1998.[3],,.[M].:,1998.[4],.[M].:,2002.[5].[M].:,2004.12()34