Some results on conditionally uniformly strong mix

J.KoreanMath.Soc.51(2014),No.3,pp.609{633⃝2014KoreanMathematicalSocietyJ.KoreanMath.Soc.51(2014),No.3,pp.609–633finitionsandsomeofbasicpropertiesofconditionallyuniformlystrongmixingran-domvariablesarederived,andseveralconditionalcovarianceinequalitiesareobtained.Bymeansofthesepropertiesandconditionalcovarianceinequalities,aconditionalcentrallimittheoremstatedintermsofcondi-tionalcharacteristicfunctionsisestablished,whichisaconditionalversionoftheearlierresultunderthenon-conditionalcase.1.IntroductionanddefinitionWewillbeworkingonafixedprobabilityspace(Ω,A,P).Considerasequence{Xn,n≥1}ofrandomvariablesandletAk1=σ(X1,...,Xk),A∞k+n=σ(Xk+n,Xk+n+1,...)betheσ-algebrasinducedbytherespectiverandomvariables,wherekandnarearbitrarypositiveintegers.Then{Xn,n≥1}issaidtobeuniformlystrongmixingorϕ-mixingifthereexistsanonnegativesequenceϕ(n)convergingtozeroasn→∞suchthat(1.1)PBAk1
−P(B)≤ϕ(n)ReceivedNovember5,2013;RevisedJanuary28,2014.2010MathematicsSubjectClassification.60E10,60E15,60G10.Keywordsandphrases.conditionallyuniformlystrongmixing,conditionalcovarianceinequality,conditionalindependence,conditionalstationarity,conditionalcentrallimittheo-rem,conditionalcharacteristicfunction.TheauthorswouldliketothanktheanonymousrefereessincerelyfortheirvaluablecommentsandimprovingtheEnglishexpressions,whichresultedinthepresentversionofthispaper.ThisworkwassupportedbyNationalNaturalScienceFoundationofChina(No.11101452),NaturalScienceFoundationProjectofCQCSTCofChina(Nos.2011BB0105,2012jjA00035)andNaturalScienceFoundationProjectofCTBUofChina(No.1352001).c2014KoreanMathematicalSociety609610D.-M.YUAN,X.-M.HU,ANDB.TAOforallB∈A∞k+nwheneverk≥1,n≥1.ThisconceptwasproposedbyIbragi-mov[7]andcondition(1.1)isessentiallytheextensiontoarbitraryprocessesoftheergodicitycoefficientintroducedbyDobrushin[3,4]forMarkovprocesses.Anequivalentwayofwriting(1.1),duetoIbragimov[8],isthat(1.2)|P(AB)−P(A)P(B)|≤ϕ(n)P(A)foreverypairofA∈Ak1andB∈A∞k+nwheneverk≥1,n≥1.Foruniformlystrongmixingrandomvariables,manysharpandelegantre-sultsareavailableinliterature,includingChenandWang[1]forcompletemomentconvergence,Utev[19]foracentrallimittheorem,Peligrad[13]foraweakinvarianceprinciple,Sen[16]forweakconvergenceofempiricalprocesses,Shao[17]foranalmostsureinvarianceprinciple,HuandWang[6]foralargedeviationprinciple,Kuczmaszewska[10]forastronglawoflargenumbers,Szewczak[18]foraMarcinkiewiczlaw,andthelike.MotivatedbyPrakasaRao[14]extendingthenotionofstrongmixingtothatofconditionallystrongmixing,furtherworkrelatedtowhichcanbefoundinYuanandLei[21],wewillnowconsideranewkindofmixingcalledcon-ditionallyuniformlystrongmixing,whichisanextensiontotheabovenon-conditionalcase.Definition1.1.LetFbeasub-σ-algebraofA.Asequence{Xn,n≥1}ofrandomvariablesiscalledconditionallyuniformlystrongmixingorcondition-allyϕ-mixinggivenF(F-uniformlystrongmixing,inshort)ifthereexistsanonnegativeF-measurablesequenceofrandomvariablesϕF(n)convergingtozeroalmostsurelyasn→∞suchthat(1.3)PBAk1∨F
−P(B|F)≤ϕF(n)a.s.foreveryB∈A∞k+nwheneverk≥1,n≥1,whereAk1∨Fdenotestheσ-algebrageneratedbyAk1∪F.TheessencebehindthisdefinitionisthattheserandomvariablesinvolvedtendtobeasymptoticallyF-independentastheygetfurtherandfurtherapart.Thiskindofprocessiscertainlyworthstudyingnotonlyforitsprobabilisticinterest,butalsobecauseofthepotentialimportanceinstatisticalapplications.Thesequence{Xn}isautomaticallyF-uniformlystrongmixingwithϕF(n)≡0,ofcourse,provideditisF-independent.Aswith(1.1)beingequivalentto(1.2),anequivalentway(seeProposition3.2below)toexpress(1.3)is(1.4)|P(A∩B|F)−P(A|F)P(B|F)|≤ϕF(n)P(A|F)a.s.foreachchoiceofeventsA∈Ak1andB∈A∞k+nwheneverk≥1,n≥1,which,willbeusedinSection2,isthekeytomanyofourconstructionsofF-uniformlystrongmixingsequences.Forallk≥1andn≥1,itfollowsthatA∞k+n⊃A∞k+n+1.Inordertoavoidbeingdistractedfromthemainpath,weassumefromnowon,withoutfurtherCONDITIONALCENTRALLIMITTHEOREMS611explicitmentioning,thatthesequence{ϕF(n)}ofmixingcoefficientsisalmostsurelymonotonicallynon-increasing.Conditionallyuniformlystrongmixingreducestotheordinary(uncondition-ally)uniformlystrongmixingiftheconditionalσ-algebraistakenas{Ø,Ω}.Conditionallyuniformlystrongmixingmayappear,atafirstglance,tobesyn-chronizedwithuniformlystrongmixing.However,thisisnotthecasebecauseExamples2.1and2.2belowshowthatuniformlystrongmixingneitherim-pl