Zero biasing and a discrete central limit theorem

arXiv:math/0509444v2[math.PR]22Nov2006TheAnnalsofProbability2006,Vol.34,No.5,1782–1806DOI:10.1214/009117906000000250cInstituteofMathematicalStatistics,2006ZEROBIASINGANDADISCRETECENTRALLIMITTHEOREMByLarryGoldsteinandAihuaXiaUniversityofSouthernCaliforniaandUniversityofMelbourneWeintroduceanewfamilyofdistributionstoapproximateP(W∈A)forA⊂{...,−2,−1,0,1,2,...}andWasumofindependentinteger-valuedrandomvariablesξ1,ξ2,...,ξnwithfinitesecondmo-ments,where,withlargeprobability,Wisnotconcentratedonalatticeofspangreaterthan1.Thewell-knownBerry–Esseentheo-remstatesthat,forZanormalrandomvariablewithmeanE(W)andvarianceVar(W),P(Z∈A)providesagoodapproximationtoP(W∈A)forAoftheform(−∞,x].However,formoregeneralA,suchasthesetofallevennumbers,thenormalapproximationbe-comesunsatisfactoryanditisdesirabletohaveanappropriatedis-crete,nonnormaldistributionwhichapproximatesWintotalvaria-tion,andadiscreteversionoftheBerry–Esseentheoremtoboundtheerror.Inthispaper,usingtheconceptofzerobiasingfordiscreterandomvariables(cf.GoldsteinandReinert[J.Theoret.Probab.18(2005)237–260]),weintroduceanewfamilyofdiscretedistributionsandprovideadiscreteversionoftheBerry–EsseentheoremshowinghowmembersofthefamilyapproximatethedistributionofasumWofinteger-valuedvariablesintotalvariation.1.Introduction.Weintroduceanewfamilyofdistributionstoapproxi-mateP(W∈A)forAasubsetofZ={...,−2,−1,0,1,2,...}andWasumofindependentinteger-valuedrandomvariablesξ1,ξ2,...,ξnwithfinitesec-ondmoments,wheretheprobabilitythatWisnotconcentratedonalatticeofspangreaterthan1islarge.WhenAisoftheform(−∞,x]andξi’shavefinitethirdmoments,wecanusethewell-knownBerry–Esseentheorem([7]and[15])whichstatesthatthereexistsanabsoluteconstantCsuchthatsupz∈RPW−μσ≤z−Φ(z)≤Cσ3nXi=1E[|ξi−E(ξi)|3],ReceivedOctober2004;revisedAugust2005.AMS2000subjectclassifications.Primary60F05;secondary60G50.Keywordsandphrases.Stein’smethod,integer-valuedrandomvariables,totalvaria-tion.ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofProbability,2006,Vol.34,No.5,1782–1806.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12L.GOLDSTEINANDA.XIAwhereμ=E(W),σ2=Var(W)andΦisthecumulativedistributionfunctionofthestandardnormal.Ifξi’sareidenticallydistributed,thentheboundisoftheordern−1/2,whichisknowntobethebestpossible.However,formoregeneralA,suchasthesetofallevennumbers,theerrorsofnormalapproximationmaybelarge,ordifficulttocompute;forsuchcases,itisde-sirabletohaveadistributionwhichapproximatesWintotalvariation,andadiscreteversionoftheBerry–Esseentheoremtoevaluatetheerror.More-over,approximationsintotalvariationhavethepropertythatanyfunctionofWisapproximatedintotalvariationtothesamedegreeasWitself,anadvantagenotenjoyedbytheKolmogorovdistance.Afewdiscretedistributions,suchassignedcompoundPoissonmeasuresandtranslatedPoissondistributions(see[6,9]andreferencestherein)havebeenproposedtomakeverycloseapproximationsintotalvariationtothedistributionofW.TheseapproximationscanbeviewedasmodificationsofPoissonapproximationandinapplications,oneoftentransformsthesumWintoaformwhichcanbeapproximatedreasonablywellbyasuitablychosenPoissonrandomvariable.Inestimatingtheerrorsofapproximation,besidestheassumptionthatWhaslargeprobabilityofnotbeingconcentratedonalatticeofspangreaterthan1,onealsoneedsotherassumptions,suchasexistenceofthethirdmomentsoftheξi’s([6],Theorem4.3),andmayadditionallyintroducetruncation.AnotherapproachistodefineadiscretenormalYbyP(Y=j)=P(j−1/2Z≤j+1/2),Z∼N(μ,σ2),j∈Z(L.H.Y.Chen,personalcommunication),thoughitisnotclearwhatqualityofapproximationYcanachieve.InthispaperweproposeaclassofapproximatingdistributionswhichhavecarrierspaceZ,thusavoidingtruncationandintegerizationproblems.Thesenewdistributionsareuniquelydeterminedbyparametersμandσ2,similarlytohowtheapproximatingnormaldistributionisdeterminedintheclassicalcentrallimittheorem.ItisexpectedthatanysuchapproximatingfamilyofdiscretedistributionsberelatedtothePoisson,adistributioncharacterizedbythepropertyofbeingequaltoitsownreducedPalmdistribution;see[23],page93.AsthispropertyisintrinsicinthestudyofcertainPoissonapproximations[1,11],andsincethePalmdistributioninvolvesonlythefirstmomentofthedistribution,itisofinteresttodeterminewhetherthereexistsanycounterparttothePoissonalsoinvolvingthesecondmoment,whichgivesadditionalflexibilityinapproximation.Oneappropriatecoun-terpartcanbeuncoveredthroughtheconceptofzerobiasing[20].Basedonthecontinuousnormalcase,itisexpectedthattheclassofapproximatingdistributionsshouldarrivenaturallyastheuniquecandidateswhichequaltheirzero-biaseddistribution.However,becauseofthediscretesetting,someZEROBIASINGANDADISCRETECENTRALLIMITTHEOREM3adjustmentsarefirstneededtomaketheideawork.InSection2,wepro-videsomebackgroundonzerobiasinginboththecontinuousanddiscretesettings,anddefineourapproximatingfamilyofdistributionsthroughamodifiedzerobiasingform.Inparticular,ourdistributionsarerelatedtotheoperator(2.11),connectedtodiscretezerobiasing,andarethestation-arylawsoftheprocesseswithcorrespondinggenerato