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1、ActaApplMath(2008)103:221234DOI10.1007/s10440-008-9232-4MeanConvergenceTheoremsforWeightedSumsofArraysofResiduallyh-integrableRandomVariablesConcerningtheWeightsunderDependenceAssumptionsDemeiYuan·BaoTaoReceived:26October2007/Accepted:17March2008/Publishedonline
2、:28March2008©SpringerScience+BusinessMediaB.V.2008AbstractFromtheclassicalnotionofuniformintegrabilityofasequenceofrandomvari-ables,anewconceptcalledresidualh-integrabilityisintroducedforanarrayofrandomvariables,concerninganarrayofconstants,whichisweakerthanothe
3、rpreviousrelatednotionsofintegrability.Martingaledifference,pairwisenegativequadrantdependence,tailφ-mixingpropertyandLp-mixingalearefourspecialkindsofdependencestructures,where1≤p≤2.Byre-latingtheresidualh-integrabilitywithsuchthesedependenceassumptions,somecon
4、ditionsareformulatedunderwhichmeanconvergencetheoremsforweightedsumsofarraysofrandomvariablesareestablished,andmanyearlierresultsareexplainedasthespecialcasesoftheonesappearinginourpresentwork.KeywordsMartingaledifferencearray·Negativequadrantdependence·Tailφ-mi
5、xingarray·Lp-mixingalearray·ResidualCesàroα-integrability·Residualh-integrabilityMathematicsSubjectClassification(2000)60F15·60F051IntroductionandDefinitionLet{Xn,n≥1}denoteasequenceofrandomvariablesonaprobabilityspace(,F,P).Fromtheclassicalnotionofuniformintegra
6、bility,Chandra[4]proposedthenotionofuni-formintegrabilityintheCesàrosenseorwhathasnowcometobeknownasCesàrouniformintegrability(CUI),throughtheconditionn−1limsupnE
7、Xk
8、I(
9、Xk
10、>a)=0.a→∞n≥1k=1D.Yuan()·B.TaoSchoolofMathematicsandStatistics,ChongqingTechnologyandBusi
11、nessUniversity,Chongqing,Chinae-mail:yuandemei@163.comB.Taoe-mail:taobao@ctbu.edu.cn222D.Yuan,B.TaoHereandthroughoutthispaperI(·)denotestheindicatorfunction.ThenotionofCUIwasextendedtoCesàroα-integrabilitybyChandraandGoswami[5]inthefollowingway:Letα∈(0,∞),aseque
12、nce{Xn,n≥1}ofrandomvariablesissaidtobeCesàroα-integrable(CI(α))ifnn−1−1αsupnE
13、Xk
14、<∞andlimnE
15、Xk
16、I(
17、Xk
18、>k)=0.n≥1n→∞k=1k=1Recently,anewkindofuniformintegrabilitycalled
