The maximum on a random time interval of a random

TheAnnalsofAppliedProbability2003,Vol.13,No.1,37–53THEMAXIMUMONARANDOMTIMEINTERVALOFARANDOMWALKWITHLONG-TAILEDINCREMENTSANDNEGATIVEDRIFT1BYSERGUEIFOSSANDSTANZACHARYHeriot-WattUniversityWestudytheasymptoticsforthemaximumonarandomtimeintervalofarandomwalkwithalong-taileddistributionofitsincrementsandnegativedrift.WeextendtoageneralstoppingtimearesultbyAsmussen,simplifyitsproofandgivesomeconverses.1.Introduction.Randomwalkswithlong-tailedincrementshavemanyimportantapplicationsininsurance,finance,queueingnetworks,storageprocessesandthestudyofextremeeventsinnatureandelsewhere.See,forexample,Embrechts,KlüppelbergandMikosch(1997),Asmussen(1998,1999)andGreiner,JobmannandKlüppelberg(1999)forsomebackground.Inthispaperwestudythedistributionofthemaximumofsucharandomwalkoverarandomtimeinterval.LetFbethedistributionfunctionoftheincrementsofarandomwalk{Sn}n≥0withS0=0.SupposethatthisdistributionhasafinitenegativemeanandthatFislong-tailedinthepositivedirection(seebelowforthisandotherdefinitions).Ofinterestistheasymptoticdistributionofthemaximumof{Sn}overtheinterval[0,σ]definedbysomestoppingtimeσ.Someresultsforthecasewhereσisindependentof{Sn}areknown(againseebelow).However,relativelylittleisknownforotherstoppingtimes.Asmussen(1998)gavetheexpectedresultforthecaseσ=τ,whereτ=min{n≥1:Sn≤0}(1)[seealsoHeath,ResnickandSamorodnitsky(1997)andGreiner,JobmannandKlüppelberg(1999)].ThisresultrequiresthefurtherconditionthatthedistributionfunctionFhavearighttailwhichbelongstotheclassS∗introducedbyKlüppelberg(1988)(weshallsimplywriteF∈S∗).InthepresentpaperweextendAsmussen’sresulttoageneralstoppingtimeσ.IndoingsowealsosimplifythederivationoftheoriginalresultandweshowthattheconditionF∈S∗isnecessaryaswellassufficientforittohold.WealsogiveausefulcharacterizationoftheclassS∗.Finally,asacorollaryofourresults,wegiveaprobabilisticproofoftheknownresultthatanydistributionfunctionG∈S∗issubexponential.ReceivedSeptember2001;revisedMay2002.1SupportedinpartbyINTASGrant265andbyEPSRCGrant58765.AMS2000subjectclassifications.Primary60G70;secondary60K30,60K25.Keywordsandphrases.Ruinprobability,long-taileddistribution,subexponentialdistribution.3738S.FOSSANDS.ZACHARYThus,let{ξn}n≥1beasequenceofindependentidenticallydistributedrandomvariableswithdistributionfunctionF.WeassumethroughoutthatEξn=−m0.(NEG)WefurtherassumethroughoutthatthedistributionfunctionFislong-tailed(LT),thatis,thatF(x)0forallx,limx→∞F(x−h)F(x)=1forallfixedh0.(LT)Here,foranydistributionfunctionGonR,GdenotesthetaildistributiongivenbyG(x)=1−G(x).Definetherandomwalk{Sn}n≥0byS0=0,Sn=
ni=1ξiforn≥1.Forn≥0,letMn=max0≤i≤nSiandletM=supn≥0Sn.Similarly,foranystoppingtimeσ(withrespecttoanyfiltration{Fn}n≥1suchthat,foreachn,ξnismeasurablewithrespecttoFnandξn+1isindependentofFn),letMσ=max0≤i≤σSi.WeareinterestedintheasymptoticdistributionofMσforageneralstoppingtimeσ(whichneednotbea.s.finite).Inparticularweareinterestedinobtainingconditionsunderwhichlimx→∞P(Mσx)F(x)=Eσ.(2)Werequirefirstsomefurtherdefinitions.ForanydistributionfunctionGonRdefinetheintegrated,orsecond-tail,distributionfunctionGsbyGs(x)=min(1,∞xG(t)dt).AdistributionfunctionGonR+issubexponentialifandonlyifG(x)0forallxandlimx→∞G∗2(x)/G(x)=2(whereG∗2istheconvolutionofGwithitself).Moregenerally,adistributionfunctionGonRissubexponentialifandonlyifG+issubexponential,whereG+=GIR+andIR+istheindicatorfunctionofR+.Itisknownthatthesubexponentialityofadistributiondependsonlyonits(right)tail,andthatasubexponentialdistributionislong-tailed.WhenFissubexponential,itiselementarythattheresult(2)holdsforanya.s.constantσ.[Thecondition(NEG)isnotrequiredhere.See,e.g.,Embrechts,KlüppelbergandMikosch(1997)orSigman(1999).]InthecasewhereFsissubexponential,theasymptoticdistributionofMisknown—inparticular,P(Mx)=O(Fs(x))asx→∞[seeVeraverbeke(1977),EmbrechtsandVeraverbeke(1982)and,forasimplertreatment,Embrechts,KlüppelbergandMikosch(1997)].AdistributionfunctionGonRbelongstotheclassS∗ifandonlyifG(x)0forallxandx0G(x−y)G(y)dy∼2mG+G(x)asx→∞,(3)wheremG+=∞0G(x)dxisthemeanofG+.ItisagainknownthatthepropertyG∈S∗dependsonlyonthetailofGandthatifG∈S∗,thenbothGandGsaresubexponential;seeKlüppelberg(1988).MAXIMUMONARANDOMTIMEINTERVAL39Considerfirstthecasewherethestoppingtimeσisindependentofthesequence{ξn}.Here,underthefurtherconditionthatthedistributionfunctionFissubexponential,theresult(2)iswellknowntoholdforanystoppingtimeσsuchthatEexpλσ∞forsomeλ0.(4)Inthisparticularcasethecondition(NEG)isnotrequired[see,e.g.,Embrechts,KlüppelbergandMikosch(1997)andreferencestherein].Thecondition(4)maybedroppedbysuitablystrengtheningthesubexponentialityconditiononF[seeBorovkovandBorovkov(2001)andKorshunov(2001)].Thefirstresultsforastoppingtimeσwhichisnotindependentofthese-quence{ξn}weregivenbyHeath,ResnickandSamorodnitsky[(1997),Proposi-tion2.1]and,undermoregeneralconditions,byAsmussen[(1998),Theorem2.1];seealsoGreiner,JobmannandKlüppelberg[(1999),Theorem3.3].Asmussenshowedthatif,inadditiontoourpresentconditions(NEG)and(LT),wehaveF∈S∗,thentheresult(2)holdswithσ=τ,whereτisasgivenby(1).[Asmussenfailedtostateform