Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications.pdf

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1、ScienceinChinaSeriesA:MathematicsSep.,2009,Vol.52,No.9,18871904www.scichina.commath.scichina.comwww.springerlink.comRosenthaltypeinequalitiesforasymptoticallyalmostnegativelyassociatedrandomvariablesandapplicationsYUANDeMei†&ANJunSchoolofMathematicsandStatistics,ChongqingTechnologyand

2、BusinessUniversity,Chongqing400067,China(email:yuandemei@163.com,scottan@sina.com)AbstractInthispaper,weestablishsomeRosenthaltypeinequalitiesformaximumpartialsumsofasymptoticallyalmostnegativelyassociatedrandomvariables,whichextendthecorrespondingresultsfornegativelyassociatedrandomv

3、ariables.Asapplicationsoftheseinequalities,byemployingthenotionsofresidualCes`aroα-integrabilityandstrongresidualCes`aroα-integrability,wederivesomeresultsonLpconvergencewhere1

4、lsumsofidenticallydistributedrandomvariables.Keywords:Rosenthaltypeinequality,asymptoticallyalmostnegativeassociation,residualCes`a-roα-integrability,strongresidualCes`aroα-integrability,Lp-convergence,completeconvergence,Marcinkiewicz-ZygmundstronglawMSC(2000):60F05,60F151Introductio

5、nSincetheconceptofnegativeassociationwasintroducedbyJoag-DevandProschan[1]andBlocketal.[2],itslimitpropertieshavearousedwideinterestbecauseoftheirnumerousapplicationsinreliabilitytheory,percolationtheoryandmultivariatestatisticalanalysis.Afinitefamilyofrandomvariables{Xi,1in}issaidto

6、benegativelyassociated(NA,inshort)ifforeverypairofdisjointsubsetsAandBof{1,2,...,n},Covf(Xi,i∈A),g(Xj,j∈B)0,wheneverfandgarecoordinatewisenondecreasingandthecovarianceexists.AninfinitefamilyisNAifeveryfinitesubfamilyisNA.ByinspectingtheproofofMatula’s[3]maximalinequalityfortheNArando

7、mvariables,onealsocanallownegativecorrelationsprovidedtheyaresmall.Primarilymotivatedbythis,ChandraandGhosal[4,5]introducedthefollowingdependence:Asequence{Xn,n1}ofrandomvariablesiscalledasymptoticallyalmostnegativelyassociated(AANA,inshort)ifthereexistsanonnegativesequenceq(n)→0asn→

8、∞such