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1、J.KoreanMath.Soc.51(2014),No.1,pp.1–15http://dx.doi.org/10.4134/JKMS.2014.51.1.001CONDITIONALCENTRALLIMITTHEOREMSFORASEQUENCEOFCONDITIONALINDEPENDENTRANDOMVARIABLESDe-MeiYuan,Li-RanWei,andLanLeiAbstract.Aconditionalversionoftheclassicalcentrallimittheoremisderivedrigorouslyby
2、usingconditionalcharacteristicfunctions,andamoregeneralversionofconditionalcentrallimittheoremforthecaseofconditionallyindependentbutnotnecessarilyconditionallyidenticallydistributedrandomvariablesisestablished.Thesearedoneanticipatingthatthefieldofconditionallimittheorywillpr
3、ovetobeofsignificantapplicability.1.IntroductionLet(Ω,A,P)beaprobabilityspaceandletFbeasub-σ-algebraofA.Afinitesequenceofrandomvariables{Xk,1≤k≤n}issaidtobeconditionallyindependentwithrespecttoF(F-independent,inshort)if,foreveryBk∈B(theBorelσ-algebrainR),nnYFF(1.1)P∩(Xk∈Bk)=P
4、(Xk∈Bk)a.s.k=1k=1Hereandinthesequel,PF(A)denotestheconditionalprobabilityofaneventA∈ArelativetoF.Aninfinitesequence{Xn,n≥1}issaidtobeF-independentifeveryfinitesubsequenceisF-independent.Notethatanequiv-alentconditionto(1.1),whichwasprovedbyRoussas[8],isnnYPF(X≤x)=PF(X≤x)a.s.∩
5、kkkkk=1k=1forevery(x,x,...,x)∈Rn.12nReceivedMarch1,2012;RevisedApril8,2013.2010MathematicsSubjectClassification.60F05,60E10.Keywordsandphrases.conditionalindependence,conditionalidenticaldistribution,con-ditionalcharacteristicfunction,conditionalcentrallimittheorem.Thisworkwas
6、supportedbyNationalNaturalScienceFoundationofChina(No.11101452),NaturalScienceFoundationProjectofCQCSTCofChina(Nos.2011BB0105,2012jjA00035)andtheSCRofChongqingMunicipalEducationCommission(No.KJ120731).c2014TheKoreanMathematicalSociety12DE-MEIYUAN,LI-RANWEI,ANDLANLEIIfF={Ω,Ø},
7、thenF-independencereducestotheordinary(uncondi-tional)independence.InPrakasaRao[7],concreteexamplesweregiven,whereindependentrandomvariableslosetheirindependenceunderconditioning,anddependentrandomvariablesbecomeindependentunderconditioning.Therandomnatureofmanyproblemsarisin
8、gintheappliedsciencesleadstomathematicalmodelswhereconditioningispre
