Weak convergence of the tail empirical process for

WeakconvergenceofthetailempiricalprocessfordependentsequencesHolgerRootz¶enChalmersUniversityofTechnologyyAbstractThispaperprovesweakconvergenceinDofthetailempiricalprocess-therenormalizedextremetailofoftheempiricalprocess-foralargeclassofstationarysequences.Theconditionsneededforconvergenceare(i)momentrestrictionsontheamountofclusteringofextremes,(ii)re-strictionsonlongrangedependence(absoluteregularityorstrongmixing),and(iii)convergenceofthecovariancefunction.Wefurthershowhowthelimitprocessischangedifexceedancesofanonrandomlevelarereplacedbyexceedancesofahighquantileoftheobservations.Weakconvergenceofthetailempiricalprocessisonekeytoasymptoticsforextremevaluestatisticsanditswiderangeofapplications,fromgeoscienceto¯nance.Earlier(unpublished)versionsofourresultshavealreadyfoundsigni¯-cantapplicationinestimationtheory.Theyalsogiveatheoreticalbasisforpopulardiagnosticplotsindependentcases.1IntroductionThispapergivesanumberofconvergenceresultsforthetailempiricalfunctionfordependentstationarysequences.Inadditiontotheoreticalinterest,themo-tivationcomesfromsemiparametricmethodsforextremes,suchasthePeaksoverThresholds(PoT)method.Inthesethestatisticalanalysisonlyusesthepartoftheobservationswhichexceedsomesuitablychosenhighlevel.Themethods,inparticularwithageneralizedParetoassumptionforthetaildistri-bution,are¯ndingsigni¯cantapplication,andaregettinga¯rmertheoreticalfoundation.Thereareseveralrecentbooksonthesubject((Coles(2001),Em-brechtsetal.(1997),KotzandNadarajah(2000),Kowaka(1994),Beirlantetal.(2005),ReissandThomas(2005))andalargejournalliterature.Theliteratureiscomplementedbyaconsiderablebodyofsoftware:forareviewseeStephensonandGilleland(2005).Keywordsandphrases:extremes,clusteringofextremes,taildistributionfunction,abso-luteregularity,strongmixing.AMS2000Classi¯cation:Primary60G70;Secondary60F17,62G32.ySE-41962GÄoteborg,SWEDEN,[e-mail]tbytheSwedishFoundationforStrategicResearch.1Dependentobservationsareofbasicinterestinmanyclassicalapplicationareasforextremevaluestatistics,e.g.GeoscienceandEnvironmentalScience.Also,intherecentsurgeofinterestinextremevaluesof¯nancialdata,depen-denceistheruleratherthantheexception.Ourlimittheoremsprovideatheoreticalfoundationforextremevaluees-timationtheory,andforgraphicaldiagnosticsbye.g.probabilityandquantileplots.Resultsfromanearlierunpublishedversion(Rootz¶en(1995))alreadyhasfoundsigni¯cantapplicationinderivingasymptoticnormalityforPoTes-timatorsindependentcases,seeDrees(2000,2002,2003).Thelatterofthesepapersalsoshowsthatneglectingdependencecanleadtosevereunderestima-tionofvariability.BesidesResnickandSt¸aric¸a(1998)whostudyestimationforsomespeci¯cheavy-tailedmodels,theonlyotherextremevalueestimationresultsfordependentsequencesareaimedatHillorHill-likeestimators(Hs-ing(1991),Rootz¶enetal(1992),ResnickandSt¸aric¸a(1995,1997),DattaandMcCormick(1998),Novak(2002),Hill(2006)).Todescribetheresultsofthispaper,letf»ig1i=¡1beastationarysequencewithcontinuousmarginaldistributionfunction(d.f.)F.Wethroughoutusethenotation¹F(x)=1¡F(x)forthetaild.f.Letfung1n=1be(high)levelsandf¾n0g1n=1benormingconstants.Thetailfunction(or\conditionaltaildistributionfunction)isde¯nedtobeTn(x)=¹F(un+x¾n)¹F(un);x¸0;n=1;2;::::InmuchofthispaperweassumethatthetailfunctionconvergestoageneralizedParetoform,i.e.,Tn(x)!T(x)=(1+°x¾)¡1=°+;x¸0;n!1;(1.1)where¾0and°2(¡1;1)areparametersofthelimitandthesubscript+denotespositivepart.For°=0weinterpretT(x)tobethelimit,e¡x=¾,as°!0.ThespecialcaseofuniformdistributionsisconsideredseparatelyinSection5,foruseasatechnicaltoolhere.ItcanalsobeusefulforsituationswheretheGeneralizedParetoassumptionisn'tsatis¯ed.LetxTbetherightendpointofthesupportofT,i.e.,xT=supfx;T(x)1gsothatxT=1for°¸0,andxT=¾=j°jfor°0.Thetailempiricaldistributionfunctionandthetailempiricalprocessarede¯nedas~Tn(x)=1n¹F(un)nXi=11f»iun+x¾ngande(~Tn)(x)=qn¹F(un)(~Tn(x)¡Tn(x));respectively,with1fgdenotingtheindicatorfunctionwhichisoneiftheeventincurlybracketsoccurs,andzerootherwise.LetD(I)bethespaceofrightcontinuousrealfunctionsonthe(¯niteorin¯nite)intervalIwhicharerightcontinuousandhaveleftlimitsateachpoint,2giventheLindvall-StoneextensionoftheShorokhodJ1-topology,seePollard(1984).The¯rstresultsofthispaper,Theorems2.1and2.2,arethatthetailempiricalprocessconvergesinD([0;xT))toacontinuousGaussianprocess.Theresultrequiresthreekindsofassumptionsbeyond(1.1).Wenowbrie°ydiscussthese.AmoredetaileddiscussionisgiveninSection4.Oftenextremesindependentsequencescomeinsmallclusters(incontrasttoindependentsequenceswhereextremestendtobeisolatedfromonean-other).The¯rstkindofassumptionrestrictsthesizeofclustersoflargevaluesbyassumingthattheyhaveasuitablyboundedp-thmoment.Thesecondas-sumptionmakesclusterswhicharefarapartasymptoticallyindependent.Forthislongrangedependencerestrictionweuseeitherabsoluteregularity(some-timesalsocalled¯-mixing)orthesomewhatweakerstrongmixingcondition.Absoluteregularitymaybethemostnaturalassumption.Itiswidelyappli-cable,e.g.insituationswherecouplingorregenerationholds