51(2014)609–633. Some results on conditionally uniformly strong mixing sequences of random variables..pdf

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1、J.KoreanMath.Soc.51(2014),No.3,pp.609{633http://dx.doi.org/10.4134/JKMS.2014.51.3.609SOMERESULTSONCONDITIONALLYUNIFORMLYSTRONGMIXINGSEQUENCESOFRANDOMVARIABLESDe-MeiYuan,Xue-MeiHu,andBaoTaoReprintedfromtheJournaloftheKoreanMathematicalSocietyVol.51,No.3,May2014⃝c20

2、14KoreanMathematicalSocietyJ.KoreanMath.Soc.51(2014),No.3,pp.609–633http://dx.doi.org/10.4134/JKMS.2014.51.3.609SOMERESULTSONCONDITIONALLYUNIFORMLYSTRONGMIXINGSEQUENCESOFRANDOMVARIABLESDe-MeiYuan,Xue-MeiHu,andBaoTaoAbstract.Fromtheordinarynotionofuniformlystrongmi

3、xingforase-quenceofrandomvariables,anewconceptcalledconditionallyuniformlystrongmixingisproposedandtherelationbetweenuniformlystrongmix-ingandconditionallyuniformlystrongmixingisansweredbyexamples,thatis,uniformlystrongmixingneitherimpliesnorisimpliedbycondi-tiona

4、llyuniformlystrongmixing.Acoupleofequivalentdefinitionsandsomeofbasicpropertiesofconditionallyuniformlystrongmixingran-domvariablesarederived,andseveralconditionalcovarianceinequalitiesareobtained.Bymeansofthesepropertiesandconditionalcovarianceinequalities,acondit

5、ionalcentrallimittheoremstatedintermsofcondi-tionalcharacteristicfunctionsisestablished,whichisaconditionalversionoftheearlierresultunderthenon-conditionalcase.1.IntroductionanddefinitionWewillbeworkingonafixedprobabilityspace(Ω,A,P).Considerasequence{Xn,n≥1}ofrando

6、mvariablesandletAk=σ(X,...,X),A∞=σ(X,X,...)11kk+nk+nk+n+1betheσ-algebrasinducedbytherespectiverandomvariables,wherekandnarearbitrarypositiveintegers.Then{Xn,n≥1}issaidtobeuniformlystrongmixingorϕ-mixingifthereexistsanonnegativesequenceϕ(n)convergingtozeroasn→∞such

7、that
(1.1)PBAk−P(B)≤ϕ(n)1ReceivedNovember5,2013;RevisedJanuary28,2014.2010MathematicsSubjectClassification.60E10,60E15,60G10.Keywordsandphrases.conditionallyuniformlystrongmixing,conditionalcovarianceinequality,conditionalindependence,conditionalstationarity,condi

8、tionalcentrallimittheo-rem,conditionalcharacteristicfunction.Theauthorswouldliketothanktheanonymousrefereessincerelyfortheirvaluablecommentsandimproving