Running Headline A Central Limit Theorem for Marti

ACentralLimitTheoremforLocalMartingaleswithApplicationstotheAnalysisofLongitudinalDataS.A.MURPHYSeptember20,1996DepartmentofStatisticsPennsylvaniaStateUniversitySUMMARYAfunctionalcentrallimittheoremforalocalsquareintegrablemartingalewithpersistentdisconti-nuitiesisgiven.Bypersistentdiscontinuities,itismeantthatthemartingalehasjumpswhichdonotvanishasymptotically.Thiscentrallimittheoremismotivatedbyproblemsintheanalysisoflongitudinalandlifehistorydata.RunningHeadline:ACentralLimitTheoremforMartingalesKeywords:LongitudinalData,EventHistoryAnalysis,Non-ClassicalCentralLimitTheorem,Mar-tingaleResearchsupportedbyNSFgrantDMS-9307255andpartiallycarriedoutduringtheauthor’svisitwiththeEconometricsDept.,FreeUniversity,Amsterdam.11.INTRODUCTIONVerylittlerecentworkhasbeendoneonNon-ClassicalCentralLimitTheoremsinadditiontotheworkbyGill(1982)andthepaperbyLiptserandShiryaev(1983)whichislaterreformulatedinthebookbyJacodandShiryaev(1987).Thecentrallimittheoremgivenhereisbasedonthelatertwoworks,buttheconditionsgivenareamenabletoapplicationsinlifehistory/longitudinaldataanalysis.Lifehistory/longitudinaldatatypicallyinvolvesobservationofentitiesorindividualsoveraperiodoftime.Eventhoughthistypeofdatamaybethoughtofastheobservationofstochasticprocesses,thestatisticalanalysisisquitedierent.Theanalysisoflifehistorydataisbasedontheobservationofseveralormanystochasticprocesseseachoverashorttimeperiodinsteadofobservationofaveryfew(orone)stochasticprocessesoveralongperiodoftime.Thereforetheasymptoticsgivenherewillbeforthenumberofindividuals/processesincreasingwithoutbound.Bothlongitudinalandlifehistorydatacanexpressedasobservationsofmarkedpointprocesses(ArjasandHaara,1992).Theeventtimesofthepointprocessarethetimesatwhichonecollectsinformationontheindividualsandthemarksaretheinformationcollected.Asymptoticresultsforestimatorsandteststatisticsarebasedonacentrallimittheoremforanestimatingequation.Theestimatingequationmaybebasedonthederivativeofthelogofthefullorpartiallikelihood.Thisderivativeformsalocalsquareintegrablemartingaleunderintegrabilityconditions(Andersenet.al.,1993).Moregenerally,estimatingequationscanbeconstructedbyparametrizingaspectsoftheconditionaldistributionoftheinformationcollectedatatimepointgiventhepast.Theseestimatingequationsareintegralswithrespecttothemarkedpointprocessandunderintegrabilityconditionsformlocallysquareintegrablemartingales(MurphyandLi,1993).Centrallimittheorems(Rebolledo,seeAndersenet.al.,1993)forcontinuoustimemartingalesassumethattheintensityofthejumpsofthemartingaleis(asymptotically)continuous.Howeveritiseasytoenvisionthesituationinwhichoneplanstomakemeasurementsoneachindividualatregularintervals(eg.every3months)butsomeindividualsappearearlierorlaterformeasurements.Thetimeatwhichtheindividualappearscoulddependonpasthistory,i.e.anappointmentforasickerpatientmaybescheduledearlierduetohealthconcerns(doctor’scareorpatientself-selection,Gruger,KayandSchumacher,1991).Theasymptotic2analysisshouldallowforbothmeasurementstakenatrandomtimesandclumpingofmeasurements(atthe3monthintervals).Additionallyindividualsmaybelosttofollowuporcensored.Thetheorempresentedinthenextsectionwillalsoallowfordependencebetweenindividualswhichisduetothecensoringmechanismandtimedependentcovariates.Thersttheoremisgivenforalocalsquareintegrablemartingale.Nextthistheoremisspecializedtoacentrallimittheoremforintegralswithrespecttoamarkedpointprocess.Lastlymotivatingapplicationsarediscussed.Alloftheproofsareintheappendix.2.ACENTRALLIMITTHEOREMThersttheoremisthemostgeneralgivenhereandisforad-dimensionallocalsquareintegrablemartingale,MnwithMn(0)=0,denedonastochasticbasisn;IFn;fIFntgt2+;Pn.AssociatedwithMnisamarkedpointprocesswhichcountsthejumpsofMn,Mn,andrecordsthesizesofthejumpsasfollows,n(dx;dt)=Ps3Mn(s)6=0Mn(s);s(dx;dt)(x2d)whereuisaprobabilitymeasuregivingmass1tothepointu.Themarkedpointprocesshasapredictablecompensatorgivenby,n(dx;dt).Fortheprecisedenitionofthepredictablesigmaeld,IFnp,andotherterminologyseeJacodandShiryaev(Chapter2,1987).UsingthismarkedpointprocessonecandecomposeMnintoacontinuouslocalsquareintegrablemartingale,Mcn,plusthecompensatedjumps,Mn()=Mcn()+R0Rx(n(dx;dt)n(dx;dt)).Itisalwayspossibletowritenas,n(dx;dt)=Kn(dx;t)n(dt)whereKn(dx;s)isatransitionfunctionfrom+n;IFnptod;B(d)andnisapredictablenondecreasingprocess.LetJnbeasubsetofdiscontinuitiesofn.ThesewillbethepersistentjumpswhichwillcontributetothexedjumpsofthelimitingGaussianprocess.ThejumpsofnwhicharenotcontainedinJnwillbeassumedtobeasymptoticallynegligible.Theaccumulationofinformationnecessaryforasymptoticsonthecontinuouspart(andtheasymp-toticallynegligiblejumps)ofMnisformedbysummingovereversmallerintervalsintime,uncorrelatedincrementsofMn.LipsterandShiryaevavoidthedetailsofhowonemightaccumulateinformationonthepersistentjumpsbyrequiringthattheconditionaldistribution(giventhepast)ofthejumpsizesapproachanormaldistributionsucientlyfast(conditionRinLiptserandShiryaev,1983).3Inapplications,itisnecessarytogivesomethoughttohowone