2LINDSKOG,RESNICK,ANDROYor(0;1]2?Ambiguityforthechoiceofasymptoticregi - PDF document

2LINDSKOG,RESNICK,ANDROYor(0;1]2?Ambiguityforthechoiceofasymptoticregimeledtotheideaofco-ecientoftaildependence[9,10,26,27,34,43],hiddenregularvariation(hrv)[20,28,32,33,38{40]andtheconditionalextremevalue(cev)model[12{14,21,35].Duetothescalinginherentinthedenitionofregularvariation,anaturaldomainforregularlyvaryingtailsisaregionclosedunderscalarmultiplicationandusuallythedomainisaconecenteredattheorigin.CommonlyusedconesincludeR+,Rd+,orthetwosidedversionsallowingnegativevaluesthatarenaturalinnanceandeconomics.However,asarguedin[14],thereisneedforotherconesaswell,particularlywhenasymptoticindependenceorasymptoticfulldependence([41,Chapter5],[45])ispresent.Goingbeyondnitedimensionalspaces,thereisaneedforacomprehensivetheorycoveringspacessuchasR1+andfunctionspaces.Fortunatelyagoodframeworkforsuchatheoryofregularvariationonmetricspacesafterremovalofapointwascreatedin[23].Theneedtoremovemorethanapoint,perhapsaclosedsetandcertainlyaclosedcone,wasarguedin[14].Theseideasbuildonw#-convergencein[11,SectionA2.6].Thispaperhasanumberofgoals:(1)Wefollowtheleadof[23]anddevelopatheoryofregularlyvaryingmea-suresoncompleteseparablemetricspacesSwithaclosedconeCremoved.Section2developsatopologyonthespaceofmeasuresonSrCwhichareniteonregionsatpositivedistancefromC.Thistopologyallowscreationofmappingtheorems(Section2.1)thatencouragecontinuityargumentsandisdesignedtoallowsimultaneousregularvariationpropertiestoexistatdierentscalesasisconsideredinhiddenregularvariation.(2)WeapplythegeneralmaterialofSection2totwosignicantapplications.(a)InSection4wefocusonRp+andR1+,thespaceofsequenceswithnon-negativecomponents.AniidsequenceX=(X1;X2;:::)2R1+suchthatP[X1�x]isregularlyvaryinghasadistributionwhichisregularlyvaryingonR1+rC6jforanyj�1,whereC6jaresequenceswithatmostjpositivecomponents.Mappingtheorems(Section2.1)allowextensiontotheregularvariationpropertiesofS=(X1;X1+X2;X1+X2+X3;:::)inR1+minusthesetofnon-decreasingsequenceswhichareconstantafterthejthcomponent.SeeSection4.5.2.Forreasonsofsimplicityandtaste,werestrictdiscus-siontoR1+butwithmodesteort,resultscouldbeextendedtoR1.WealsodiscussregularvariationofthedistributionofasequenceofPoissonpointsinR1+(Section4.5.4).(b)TheR1+discussionofPoissonpointsinSection4.5.4canbeleveragedinanaturalwaytoconsider(Section5)regularvariationofthedis-tributionofaLevyprocesswhoseLevymeasureisregularlyvary-ing:limt!1t�b(t)x;1
=x�;x�0,forsomescalingfunctionb(t)!1.Wereproducetheresult[22,24]thatthelimitmeasureofregularvariationwithscalingb(t)onD([0;1];R)rf0gconcentratesoncadlagfunctionswithonepositivejump.ThisraisesthenaturalquestionofwhathappenedtotherestofthejumpsoftheLevyprocess 4LINDSKOG,RESNICK,ANDROYdenotetheclassofreal-valued,non-negative,boundedandcontinuousfunctionsonS,andletMbdenotetheclassofniteBorelmeasuresonS.Abasicneighborhoodof2Mbisasetoftheformf2Mb:jRfid�Rfidj;i=1;:::;kg,where&#x-278;0andfi2Cbfori=1;:::;k.Thusasub-basisforMbaresetsoftheformf2Mb:(f):=Rfd2Ggforf2CbandGopeninR+.ThisequipsMbwiththeweaktopologyandconvergencen!inMbmeansRfdn!Rfdforallf2Cb.Seee.g.Sections2and6in[6]fordetails.FixaclosedsetCSandsetO=SrC,e.g.onepossiblechoiceisO=Srfs0gforC=fs0gforsomes02S.ThesubspaceOisametricsubspaceofSintherelativetopologywith-algebraSO=S(O)=fA:AO;A2Sg.LetCO=C(O)denotethereal-valued,non-negative,boundedandcontinuousfunctionsfonOsuchthatforeachfthereexistsr&#x-278;0suchthatfvanishesonCr;weusethenotationCr=fx2S:d(x;C)rg,whered(x;C)=infy2Cd(x;y).Similarly,wewillwrited(A;C)=infx2A;y2Cd(x;y)forAS.WesaythatasetA2SOisboundedawayfromCifASrCrforsomer&#x-278;0orequivalentlyd(A;C)&#x-278;0.SoCOconsistsofnon-negativecontinuousfunctionswhosesupportsareboundedawayfromC.LetMObetheclassofBorelmeasuresonOwhoserestrictiontoSrCrisniteforeachr&#x-278;0.Whenconvenient,wealsowriteM(O)orM(SrC).Abasicneighborhoodof2MOisasetoftheformf2MO:jRfid�Rfidj;i=1;:::;kg,where&#x-348;0andfi2COfori=1;:::;k.Asub-basisisformedbysetsoftheformf2MO:(f)2Gg;f2CO;GopeninR+:(2.1)Convergencen!inMOisconvergenceinthetopologydenedbythisbaseorsub-base.For2MOandr&#x-348;0,let(r)denotetherestrictionoftoSrCr.Then(r)isniteandisuniquelydeterminedbyitsrestrictions(r),r&#x-348;0.Moreover,convergenceinMOhasanaturalcharacterizationintermsofweakconvergenceoftherestrictionstoSrCr.Theorem2.1(Portmanteautheorem).Let;n2MO.Thefollowingstate-mentsareequivalent.(i)n!inMOasn!1.(ii)Rfdn!Rfdforeachf2COwhichisalsouniformlycontinuousonS.(iii)limsupn!1n(F)6(F)andliminfn!1n(G)&#x-348;(G)forallclosedF2SOandopenG2SOandFandGareboundedawayfromC.(iv)limn!1n(A)=(A)forallA2SOboundedawayfromCwith(@A)=0.(v)(r)n!(r)inMb(SrCr)forallbutatmostcountablymanyr&#x-348;0.(vi)Thereexistsasequencefrigwithri#0suchthat(ri)n!(ri)inMb(SrCri)foreachi.Forproofs,seeSection2.4.Note,theresultistrueforanygeneralmetricspace.Weakconvergenceismetrizable(forinstancebytheProhorovmetric;seee.g.p.72in[6])andthecloserelationbetweenweakconvergenceandconvergenceinMOin 6LINDSKOG,RESNICK,ANDROY2.2.RelativecompactnessinMO.Provingconvergencesometimesrequiresacharacterizationofrelativecompactness.Asubsetofatopologicalspaceisrelativelycompactifitsclosureiscompact.Asubsetofametricspaceiscompactifandonlyifitissequentiallycompact.Hence,MMOisrelativelycompactifandonlyifeverysequencefnginMcontainsaconvergentsubsequence.For2MMOandr�0,let(r)betherestrictionoftoSrCrandM(r)=f(r):2Mg.ByTheorem2.1(vi)wehavethefollowingcharacterizationofrelativecompactness.Theorem2.4.AsubsetMMOisrelativelycompactifandonlyifthereexistsasequencefrigwithri#0suchthatM(ri)isrelativelycompactinMb(SrCri)foreachi.Prohorov'stheoremcharacterizesrelativecompactnessintheweaktopology.ThistranslatestoacharacterizationofrelativecompactnessinMO.Theorem2.5.MMOisrelativelycompactifandonlyifthereexistsasequencefrigwithri#0suchthatforeachisup2M(SrCri)1;(2.3)andforeach&#x]TJ/;༔ ; .96;& T; 10;&#x.516;&#x 0 T; [0;0thereexistsacompactsetKiSrCrisuchthatsup2M(Sr(Cri[Ki))6:(2.4)2.3.M-convergencevsvagueconvergence.Vagueconvergencecomplieswiththetopologyonthespaceofmeasureswhichareniteoncompacta.RegularvariationformeasuresonaspacesuchasRp+hastraditionallybeenformulatedusingvagueconvergenceaftercompacticationofthespace.InordertomakeuseofexistingregularvariationtheoryonRp+,itisusefultounderstandhowM-convergenceisrelatedtovagueconvergence.LetSbeacompleteseparablemetricspaceandsupposeCisclosedinS.ThenM+(SrC)isthecollectionofmeasuresniteonK(SrC),thecompactaofSrC:M+(SrC)=f:(K)1;8K2K(SrC)g:VagueconvergenceonM+(SrC)means7!(f)iscontinuousforallf2C+K(SrC),thecontinuousfunctionswithcompactsupport.ThespacesM+(SrC)andM(SrC)arenotthesame.ForexampleifS=[0;1)andC=f0g,2M(SrC)means(x;1)1forx&#x]TJ/;༔ ; .96;& T; 13;&#x.02 ;� Td;&#x [00;0but2M+(SrC)means([a;b])1for0ab1.ForinstanceLebesguemeasureisinM+(SrC)butnotinM(SrC).2.3.1.ComparingMvsM+.Wehavethefollowingcomparison.Lemma2.1.M-convergenceimpliesvagueconvergenceand(2.5)M(SrC)M+(SrC);C+K(SrC)C(SrC):Proof.Iff2C+K(SrC),itscompactsupportKSrCmustbeboundedawayfromCandhenced(K;C)&#x]TJ/;༔ ; .96;& T; 57;&#x.388;&#x 0 T; [0;0andf2C(SrC):If2M(SrC);andDsatisesd(D;C)&#x]TJ/;༔ ; .96;& T; 57;&#x.388;&#x 0 T; [0;0,then(D)1.IfK2K(SrC)thend(K;C)&#x]TJ/;༔ ; .96;& T; 10;&#x.516;&#x 0 T; [0;0andso(K)1,showingany2M(SrC)isalsoinM+(SrC): 8LINDSKOG,RESNICK,ANDROYProof.Noticerstthat@(SrC)=fx2S:d(x;C)=gso@(SrC1)\@(SrC2)=;for16=2.TheconclusionfollowsfromLemma2.4.2.4.2.ProofofTheorem2.1.Weshowthat(i))(ii),(ii))(iii),(iii))(iv),(iv))(v),(v))(vi)and(vi))(i).Supposethat(i)holds.Supposen!inMOandtakef2CO.Given�0considertheneighborhoodN;f()=f:jRfd�Rfdjg.Byassumptionthereexistsn0suchthatn&#x-368;n0impliesn2N;f(),i.e.jRfdn�Rfdj.HenceRfdn!Rfd.Supposethat(ii)holds.TakeanyclosedFthatisboundedawayfromC.Thenthereexistsr&#x-324;0suchthatFSrCr.Soforallx2F,d(x;C)&#x-324;r.SoifwedeneF=fx2S:d(x;F)g,theneachFisopen,FFandF#Fas#0.Alsoforr=2,wehavethatforallx2Fd(x;C)&#x-364;r�r=2=r=2,meaningthatFSrCr=2.For&#x-364;0,SrFisclosedandF\(SrF)=;.Sofor0r=2,byLemma2.3,thereexistsauniformlycontinuousfunctionffromSto[0;1]suchthatf0onSrFandf1onF.Observethatf2COasFSrCr=2.Sowehavelimsupn!1n(F)6limn!1Zfdn=Zfd6(F):As#0,F#FandasFisclosed,wehave(F)#(F).Thisleadstolimsupn!1n(F)6(F):NowtakeanyopenGboundedawayfromC.Thenthereexistsr&#x-304;0suchthatGSrCr.SoifwedeneG=Srfx2SrG:d(x;SrG)g,theneachGisclosed,GGandG"Gas#0.SobyLemma2.3,thereexistsauniformlycontinuousfunctionffromSto[0;1]suchthatf0onSrGandf1onG.Observethatf2COasGSrCr.Sowehaveliminfn!1n(G)&#x-283;limn!1Zfdn=Zfd&#x-283;(G):As#0,G"GandasGisopen,wehave(G)"(G).Thisleadstoliminfn!1n(G)&#x-283;(G):Thiscompletestheproofof(iii).Supposethat(iii)holdsandtakeA2SOboundedawayfromCwith(@A)=0.limsupn!1n(A)6limsupn!1n(A�)6(A�)=(A)6liminfn!1n(A)6liminfn!1n(A):Hence,limn!1n(A)=(A),sothat(iv)holds.Supposethat(iv)holdsandtaker&#x-283;0suchthat(@(SrCr))=0.ByLemma2.5,allbutatmostcountablymanyr&#x-283;0satisfythisproperty.AsSrCristriviallyboundedawayfromC,wehavethatn(SrCr)!(SrCr).NowanyASrCrisalsoboundedawayfromCandasSrCrisclosed,@SrCrA=@A,wheretherstexpressiondenotestheboundaryofAwhenconsideredasasubset 10LINDSKOG,RESNICK,ANDROYThenisameasure.Clearly,�0and(;)=0.Moreover,iscountablyadditive:fordisjointAn2SOthemonotoneconvergencetheoremimpliesthat([nAn)=limr!0r([nAn\[SrCr])=limr!0Xnr(An\[SrCr])=Xn(An):2.4.4.ProofofTheorem2.3.Firstly,Dh2SO[6,p.243].TakeA02SO0boundedawayfromC0withh�1(@A0)=0.Since@h�1(A0)h�1(@A0)[Dh(seee.g.(A2.3.2)in[11]),wehave(@h�1(A0))6h�1(@A0)+(Dh)=0.Sincen!inMO,(@h�1(A0))=0,andh�1(A0)isboundedawayfromC,itfol-lowsfromTheorem2.1(iv)thatnh�1(A)!h�1(A).Hence,nh�1!h�1inMO0.2.4.5.ProofofCorollary2.1.TakeA0S0rC0suchthatd0(A0;C0)�0.Weclaimthisimpliesd(h�1(A0);C)�0.Otherwise,ifd(h�1(A0);C)=0,thereexistxn2h�1(A0)andyn2Csuchthatd(xn;yn)!0:Thenh(xn)2A0;h(yn)2h(C)=C0andifhisuniformlycontinuous,thend0(h(xn);h(yn))!0sothatd0(A0;C0)=0,acontradiction.2.4.6.ProofofCorollary2.2.TheproofofCorollary2.1showsthatitsucesifeitherfxngorfynghasalimitpoint.Intheformercase,ifxn0!xforsomesubsequencen0!1,thend(x;yn0)!0andyn0!x2Candh(yn0)!h(x)sod0(A0;C0)=0againgivingacontradiction.NoteifSiscompactthanfxnghasalimitpoint.Ontheotherhand,iffynghasalimitpointthenthereexistsaninnitesubsequencefn0gandyn0!y2Csothatd(xn0;y)!0.Thusifhiscontinuous,h(xn0)!h(y)2h(C)=C0whichcontradictsd0(A0;C0)�0.NoteifCiscompact,thenfynghasalimitpointandinparticularifC=fs0g.Thuswehavethesecondvariant.2.4.7.ProofofTheorem2.4.SupposeMMOisrelativelycompact.LetfngbeasubsequenceinM.Thenthereexistsaconvergentsubsequencenk!forsome2M�.ByTheorem2.1(v),thereexistsasequencefrigwithri#0suchthat(ri)nk!(ri)inMb(SrCri).Hence,M(ri)isrelativelycompactinMb(SrCri)foreachsuchri.Conversely,supposethereexistsasequencefrigwithri#0suchthatM(ri)Mb(SrCri)isrelativelycompactforeachi,andletfngbeasequenceofelementsinM.Weuseadiagonalargumenttondaconvergentsubsequence.SinceM(r1)isrelativelycompactthereexistsasubsequencefn1(k)goffngsuchthat(r1)n1(k)convergestosomer1inMb(SrCr1).SimilarlysinceM(r2)isrelativelycompactandfn1(k)gMthereexistsasubsequencefn2(k)goffn1(k)gsuchthat(r2)n2(k)convergestosomer2inMb(SrCr2).Continuinglikethis;foreachi�3letni(k)beasubsequenceofni�1(k)suchthat(ri)ni(k)convergestosomeriinMb(SrCri).Thenthediagonalsequencefnk(k)gsatises(ri)nk(k)!riinMb(SrCri)foreach 12LINDSKOG,RESNICK,ANDROY(3)SetS=[0;1)fx2R2+:kxk=1gandC=f0gfx2R2+:kxk=1g.For�0,dene(;(r;a))7!(r;a):3.2.Regularvariation.Recallfrome.g.[7]thatapositivemeasurablefunctioncdenedon(0;1)isregularlyvaryingwithindex2Riflimt!1c(t)=c(t)=forall�0.Similarly,asequencefcngn�1ofpositivenumbersisregularlyvaryingwithindex2Riflimn!1c[n]=cn=forall�0.Here[n]denotestheintegerpartofn.Denition3.1.Asequencefngn�1inMOisregularlyvaryingifthereexistsanincreasingsequencefcngn�1ofpositivenumberswhichisregularlyvaryingandanonzero2MOsuchthatcnn!inMOasn!1.Thechoiceofterminologyismotivatedbythefactthatfn(A)gn�1isaregularlyvaryingsequenceforeachsetA2SOboundedawayfromC,(@A)=0and(A)�0.WewillnowdeneregularvariationforasinglemeasureinMO.Denition3.2.Ameasure2MOisregularlyvaryingifthesequencef(n
)gn�1inMOisregularlyvarying.Therearemanyequivalentformulationsofregularvariationforameasure2MO.Somearenaturalforstatisticalinference.Considerthefollowingstatements.(i)Thereexistanonzero2MOandaregularlyvaryingsequencefcngn�1ofpositivenumberssuchthatcn(n
)!(
)inMOasn!1.(ii)Thereexistanonzero2MOandaregularlyvaryingfunctioncsuchthatc(t)(t
)!(
)inMOast!1.(iii)Thereexistanonzero2MOandasetE2SOboundedawayfromCsuchthat(tE)�1(t
)!(
)inMOast!1.(iv)Thereexistanonzero2MOandanincreasingsequencefbngn�1ofpositivenumberssuchthatn(bn
)!(
)inMOasn!1.(v)Thereexistanonzero2MOandanincreasingfunctionbofsuchthatt(b(t)
)!(
)inMOast!1.Theorem3.1.Thestatements(i)-(v)areequivalentandeachstatementimpliesthatthelimitmeasurehasthehomogeneityproperty(3.1)(A)=�(A)forsome�0andallA2SOand�0.Noticethataregularlyvaryingmeasuredoesnotcorrespondtoasinglescalingparameterunlessthemultiplicationoperationwithscalarsisxed.3.3.Moreexamples.WeamplifythediscussionofSection3.1.1.3.3.1.ContinuationofSection3.1.1.Example3.1.ConsideragainthecontextofSection3.1.1,item1whereS=R2andletC=(f0gR)[(Rf0g).ConsidertwoindependentParetoran-domvariables:LetX1bePa( 1)andX2bePa( 2).Dene(;(x1;x2))7! 14LINDSKOG,RESNICK,ANDROYExample3.4(Polarcoordinates).SetS=[0;1)2;C=f0g;O=[0;1)2rf0gwithscalingfunction(;x)=(;(x1;x2))7!x=(x1;x2):Forsomechoiceofnormx7!kxkdene@=fx2S:kxk=1g:AlsodeneS0=[0;1)@;C0=f0g@;O0=(0;1)@;andscalingoperationonO0is�;(r;a)
7!(r;a):Themaph(x)=�kxk;x=kxk
fromO7!O0iscontinuous.Letdandd0bethethemetricsonSandS0.SupposeXhasaregularlyvaryingdistributiononOsothatforsomeb(t)!1,(3.4)tPX=b(t)2
!(
)inMOforsomelimitmeasure.Weshowh(X)=:(R;)hasaregularlyvaryingdistributiononO0.WeapplyTheorem2.3sosupposeA0O0satisesd0(A0;f0g@)�0;thatis,A0isboundedawayfromthedeletedportionofS0.Theninffr�0:(r;a)2A0g=�0andh�1(A0)=fx2O:�kxk;x=kxk
2A0gsatisesinffkxk:x2h�1(A0)g=0�0.SothehypothesesofTheorem2.3aresatisedandallowtheconclusionthattP�R b(t);
2
!h�1(
)inMO0:(3.5)Conversely,givenregularvariationonO0asin(3.5),deneg:O07!Obyg(r;a)=ra.Mimicthevericationabovetoconclude(3.5)implies(3.4).Example3.5.Examples3.3and3.4typifythefollowingparadigm.Considertwotriples(S;C;O)and(S0;C0;O0),andahomeomorphismh:O!O0withthepropertythath�1(A0)isboundedawayfromCifA0isboundedawayfromC0.Themultiplicationbyascalar(;x)7!xinOgivesrisetothemultiplicationbyascalar(0;x0)7!0x0:=h(0h�1(x0))inO0.Noticethat(0;x0)7!0x0iscontinuous,1x0=x0,and01(02(x0))=(0102)x0.Wealsoneedtocheckthatd0(0x0;C0)�d0(x0;C0)if0�1.3.4.Proofs.3.4.1.Preliminaries.ForA2SO,writeS(A)=fx:x2A;�1g.Lemma3.1.Let2MObenonzero.Thereexistsx2Oand�0suchthatS(Bx;)isboundedawayfromC,(S(Bx;))�0,and(@rS(Bx;))=0forr�1insomesetofpositivemeasurecontaining1.Proof.Thersttwopropertiesareobvious.Inordertoprovethenalclaim,set (r)=d(@rS(Bx;);C).Noticethat (r)=d(@rBx;;C)andthat@rS(Bx;)OrC (r).Choosex2Oand�0suchthat(@S(Bx;))=0and(@(OrC (1)))=0,andsuchthat (r0)� (1)forsomer0�1.TheexistenceofsuchxandfollowsfromLemma2.5.Lemma2.5alsoimpliesthat(@(OrC ))=0forallbutatmostcountablymany 2[ (1); (r0)].Since (r)isanondecreasingand 16LINDSKOG,RESNICK,ANDROYSupposethat(i)holdsandsetc(t)=c[t].ForeachA2Aandt�1itholdsthatc[t] c[t]+1c[t]+1(([t]+1)A)6c(t)(tA)6c[t]([t]A):(3.6)Sincefcngn�1isregularlyvaryingitholdsthatlimn!1cn=cn+1=1.Hence,limt!1c(t)(tA)=(A)forallA2A.ItfollowsfromLemma3.2that(ii)holds.Supposethat(ii)holds.Thenc[t]([t]
)!(
)inMO.Moreover,fc[t]gisaregularlyvaryingsequencesincec(t)isaregularlyvaryingfunction.Therefore,(ii)implies(i).Supposethat(ii)holds.TakeasetE2SOboundedawayfromCsuchthat(tE);(E)�0and(@E)=0.Then(t
) (tE)=c(t)(t
) c(t)(tE)!(
) (E)ast!1.Hence,byTheorem2.1(ii),(iii)holds.Supposethat(iii)holds.Itwasalreadyprovedin(a)abovethatstatement(iii)impliesthatt7!(tE)isregularlyvaryingwithindex�60.Settingc(t)=1=(tE)impliesthatc(t)isregularlyvaryingwithindexandthatc(t)(t
)!(
)inMO.Thisprovesthat(iii)implies(ii).Uptothispointwehaveprovedthatstatements(i)-(iii)areequivalent.Supposethat(iv)holds.Setb(t)=b[t]andtakeA2A.Then[t] [t+1][t+1](b[t+1]A)6t(b(t)A)6[t+1] [t][t](b[t]A)fromwhichitfollowsthatlimt!1t(b(t)A)=(A).ItfollowsfromLemma3.2that(v)holds.If(v)holds,thenitfollowsimmediatelythatalso(iv)holds.Hence,statements(iv)and(v)areequivalent.Supposethat(iv)holds.TakeEsuchthat(@E)=0and(E)�0.Fort�b1,letk=k(t)bethelargestintegerwithbk6t.Thenbk6tbk+1andk!1ast!1.Hence,forA2A,k k+1(k+1)(bk+1A) k(bkE)6(tA) (tE)6k+1 kk(bkA) (k+1)(bk+1E)fromwhichitfollowsthatlimt!1(tA)=(tE)=(A)=(E).ItfollowsfromLemma3.2that(iii)holds.Hence,eachofthestatments(iv)and(v)implieseachofthestatements(i)-(iii).Supposethat(iii)holds.Thenc(t):=1=(tE)isregularlyvaryingatinnitywithindex�0.If�0,thenc(c�1(t))tast!1byPropositionB.1.9(10)in[16]andthereforelimt!1t(c�1(t)A)=limt!1c(c�1(t))(c�1(t)A)=(A)forallA2SOboundedawayfromCwith(@A)=0.If=0,thenProposition1.3.4in[7]saysthatthereexistsacontinuousandincreasingfunctionecsuchthat 18LINDSKOG,RESNICK,ANDROYandwealsoneedd01(x;y)=1Xp=1�Ppl=1jxl�ylj
^1 2p=1Xp=1kxjp�yjpk1^1 2p;wherek
k1istheusualL1normonRp+.Proposition4.1.Themetricsd1andd01areequivalentonR1+andd1(x;y)6d01(x;y)62d1(x;y):Proof.Firstofall,d01(x;y)=1Xi=1�Pil=1jxl�ylj
^1 2i�1Xi=1jxi�yij^1 2i=d1(x;y):Fortheotherinequality,observed01(x;y)=1Xi=1�Pil=1jxl�ylj
^1 2i61Xi=1Pil=1�jxl�ylj^1
2i=1Xl=11Xi=l2�i�jxl�ylj^1
=1Xl=12
2�l�jxl�ylj^1
=2d1(x;y):4.2.Continuityofmaps.WithaviewtowardapplyingCorollary2.1,weconsiderthecontinuityofseveralmaps.4.2.1.CUMSUM.WebeginwiththemapCUMSUM:R1+7!R1+denedbyCUMSUM(x)=(x1;x1+x2;x1+x2+x3;:::):Proposition4.2.ThemapCUMSUM:R1+7!R1+isuniformlycontinuousand,infact,isLipshitzinthed1metric.Proof.Wewrited1�CUMSUM(x);CUMSUM(y)
=1Xi=1Pil=1xl�Pil=1yl^1 2i61Xi=1�Pil=1jxl�ylj
^1 2i=d01(x;y)62d1(x;y):WecannowapplyCorollary2.1.Corollary4.1.LetS=S0=R1+andsupposeCisclosedinR1+andCUMSUM(C)isclosedinR1+:Ifforn�0,n2M(R1+rC)andn!0inM(R1+rC),thennCUMSUM�1!0CUMSUM�1inM(R1+rCUMSUM(C)).Forexample,ifC=f01g,thenCUMSUM(C)=f01g.Foradditionalexamples,see(4.6). 20LINDSKOG,RESNICK,ANDROYWhenremovingmorefromthestatespacethanjustf0pg,theconventionalpolarcoordinatetransform(4.1)isnotusefulif@isnotcompact,oratleastboundedawayfromwhatisremoved.Forexample,ifSrC=(0;1)p,@isnotcompactnorboundedawayfromtheremovedaxes.Thefollowinggeneralization[14]sometimesresolvesthis,provided(4.2)belowholds.Temporarilly,weproceedgenerallyandassumeSisacomplete,separablemetricspaceandthatscalarmultiplicationisdened.IfCisacone,C=Cfor�0.SupposefurtherthatthemetriconSsatises(4.2)d(x;y)=d(x;y);�0;(x;y)2SS:Note(4.2)holdsforaBanachspacewheredistanceisdenedbyanorm.(ItdoesnotholdforR1+.)IfweintendtoremovetheclosedconeC,set@C=fs2SrC:d(s;C)=1g;whichplaystheroleoftheunitsphereandC0=f0g@Cisclosed.DenethegeneralizedtransformpolarcoordinatetransformationGPOLAR:SrC7!(0;1)@C=[0;1)@Crf0g@C=S0rC0by(4.3)GPOLAR(s)=�d(s;C);s=d(s;C)
;s2SrC:SinceCisaconeandd(
;
)hasproperty(4.2),wehaveforanys2SrCthatds d(s;C);C=ds d(s;C);1 d(s;C)C=1 d(s;C)d(s;C)=1;sothesecondcoordinateofGPOLARbelongsto@C.Forexample,ifS=R2+andweremovetheconeconsistingoftheaxesthrough02,thatis,C=f0g[0;1)[[0;1)f0g,then@C=fx2R2+:x1^x2=1g.TheinversemapGPOLAR�1:(0;1)@C7!SrCisGPOLAR�1(r;a)=ra;r2(0;1);a2@C:ItisrelativelyeasytocheckthatifA0(0;1)@CisboundedawayfromC0=f0g@C,thenGPOLAR�1(A0)isboundedawayfromC.On(0;1)@Cadoptthemetricd0�(r1;a1);(r2;a2)
=jr1�r2j_d@C(a1;a2);whered@C(a1;a2)isanappropriatemetricon@C.Supposed0(A0;f0g@C)=�0:Thismeans=inf(r1;a1)2A0a22@Cd0�(r1;a1);(0;a2)
andsettinga2=a1thisisinf(r1;a1)2D0r1.Weconcludethat(r;a)2A0impliesr�.SinceGPOLAR�1(A0)=fra:(r;a)2A0g;wehaveinSrC,rememberingthatCisassumedtobeacone,d(fra:(r;a)2A0g;C)=inf(r;a)2A0d(ra;C)=inf(r;a)2A0d(ra;rC) 22LINDSKOG,RESNICK,ANDROYProof.SupposeCsatises(4.4)and(4.5)holds.Supposef2C(R1+rC)andwith-outlossofgeneralitysupposefisuniformlycontinuouswithmodulusofcontinuity!f()=sup(x;y)2R1+rCd1(x;y)jf(x)�f(y)j:Thereexists1&#x]TJ/;༔ ; .96;& T; 15;&#x.712;&#x 17.;ࢗ ;&#xTd [;&#x]TJ/;༔ ; .96;& T; 15;&#x.712;&#x 17.;ࢗ ;&#xTd [;0suchthatd1(x;C)impliesf(x)=0.Observe,(4.7)d1�(xjp;01);x
61Xj=p+12�j=2�p:Pickanypsolargethat2�p=2anddeneg(x1;:::;xp)=f(x1;:::;xp;01):Thenwehave(a)From(4.7),jf(x)�g(xjp)j=jf(x)�f(xjp;01)j6!f(2�p):(b)g2C(Rp+rPROJp(C))andgisuniformlycontinuous.ToverifythatthesupportofgispositivedistanceawayfromPROJp(C),supposedpistheL1metriconRp+anddp�(x1;:::;xp);PROJp(C)
=2:Thenthereis(z1;:::;zp)2PROJp(C)suchthatdp�((x1;:::;xp);(z1;:::;zp)
.Butthenifz2Cwithzjp=(z1;:::;zp),wehave,since(z1;:::;zp;01)2Cby(4.4),d1�(x1;:::;xp;01);(z1;:::;zp;01)
=pXi=1jxi�zij^1 2i6pXi=1jxi�zij^16pXi=1jxi�zij=dp�(x1;:::;xp);(z1;:::;zp)
;andthereforedp�(x1;:::;xp);PROJp(C)
=2impliesg(x1;:::;xp)=f(x1;:::;xp;01)=0:SothesupportofgisboundedawayfromPROJp(C)asclaimed.Nowwriten(f)�0(f)=[n(f)�n(gPROJp)]+[n(gPROJp)�0(gPROJp)]+[0(gPROJp)�0(f)]=A+B+C:(4.8)From(4.5),sincegPROJp2C(Rp+rPROJp(C)),wehaveB=n(gPROJp)�0(gPROJp)!0asn!1:HowtocontrolA?Forx2R1+rC,ifd1�(x1;:::;xp;01);C
=2,thenf((x1;:::;xp;01)=0andalsod1�x;C)6d1(x;(xjp;01)
+d1�(xjp;01);C
2�p+=2sof(x)=0:Therefore,on=fx2R1+rC:d1�(xjp;0);C
=2g 24LINDSKOG,RESNICK,ANDROYProposition4.4.Supposeforeveryn�0thatn2M+( )andnplacesnomassonthelinesthrough1p:n(�[0;1]pr[0;1)p
\Cc)=0:(4.9)Thennv!0inM+( );(4.10)ifandonlyiftherestrictionstothespacewithoutthelinesthrough1pconverge:n0:=n(
\ 0)!0(
\ 0)=:00inM( 0):(4.11)Proof.Given(4.11),letf2C+K( ).Thentherestrictionto 0satisesfj 02C( 0)son(f)=n0(fj 0)!00(fj 0)=0(f);sonv!0inM+( ).Conversely,assume(4.10).SupposeB2S( 0)and00(@ 0B)=0,where@ 0BisthesetofboundarypointsofBin 0.Thisimplies0(@ B)=0since@ (B)@ 0B[� r 0
\C:Thereforen(B)!0(B)andbecauseof(4.9),n0(B)!00(B)whichproves(4.11).4.5.RegularvariationonRp+andR1+.Forthissection,eitherSisRp+orR1+andCisaclosedcone;thenSrCisstillacone.ApplyingDenition3.2,arandomelementXofSrChasaregularlyvaryingdistributionifforsomeregularlyvaryingfunctionb(t)!1,ast!1,tP[X=b(t)2
]!(
)inM(SrC);forsomelimitmeasure2M(SrC).InRp+,ifC=f0pgorif(4.9)holds,thisdenitionisthesameastheoneusingvagueconvergenceonthecompactiedspace.4.5.1.Theiidcase:removef01g.SupposeX=(X1;X2;:::)isiidwithnon-negativecomponents,eachofwhichhasaregularlyvaryingdistributionon(0;1)satisfyingP[X1�tx]=P[X1�t]!x�;ast!1;x�0;�0:Equivalently,ast!1,(4.12)tF(b(t)
):=tP[X1=b(t)2
]!(
)inM((0;1));where(x;1)=x�;�0:TheninM(R1+rf01g),wehavet((dx1;dx2;:::):=tP[X=b(t)2(dx1;dx2;:::)]!1Xl=1Yi6=l0(dxi)(dxl)=:(0)(dx1;dx2;:::);(4.13) 26LINDSKOG,RESNICK,ANDROYTheabovediscussioncouldhavebeencarriedoutwithminormodicationswith-outtheiidassumptionbyassuming(4.12)andP[Xj�x]=P[X1�x]!cj�0;j�2:4.5.2.Theiidcase;removemore;hiddenregularvariation.Wenowinvestigatehowtogetpasttheonebigjumpheuristicbyusinghiddenregularvariation.Forj�1,setC=j=fx2R1+:1Xi=1xi�(0;1)
=jg;C6j=fx2R1+:1Xi=1xi�(0;1)
6jg=C6(j�1)[C=j;(4.18)sothatC6jisclosed.Weimagineaninnitesequenceofreductionsofthestatespacewithscalingadjustedateachstep.Thisissuggestedbythepreviousdiscus-sion.OnM(R1+rf01g),thelimitmeasure(0)concentratedonC=1,asmallpartofthepotentialstatespace.Removef01g[C=1=C61andonM(R1+rC61)seekanewconvergenceusingadjustedscalingb(p t).WegetinM(R1+rC61)ast!1,(1)t(dx1;dx2;:::)=tP[X=b(p t)2(dx1;dx2;:::)]!(1)�(dx1;dx2;:::)
:=XlYj=2fl;kg0(dxj)(dxl)(dxk)(4.19)whichconcentratesonC=2.Ingeneral,wendthatinM(R1+rC6j)ast!1,(j)t(dx1;dx2;:::)=tP[X=b(t1=(j+1)2(dx1;dx2;:::)]!(j)�(dx1;dx2;:::)
:=Xi12


j+1Yj=2fi1;:::;ij+1g0(dxj)(dxi1)(dxi2):::(dxij+1)(4.20)whichconcentratesonC=(j+1).Thisisanelaborationofresultsin[28,31,32].TheresultinR1+canbeprovenbyreducingtoRp+bymeansofTheorem4.1notingthatC6jsatises(4.4)andthenobservingthatneither(j)tnor(j)putsmassonlinesthrough1p.Itisenoughtoshowconvergencesofthefollowingform:Assumep&#xi]TJ;&#x/F9 ;.98; T; 9.;E ;&#x-0.9;— T; [0;jandi1i2


ij+1andyl&#x-278;0;l=1;:::;j+1andtP[Xil&#x-278;b(t1=(j+1))yl;l=1;:::;j+1]=j+1Yl=1t1=(j+1)P[Xil&#x-278;b(t1=(j+1))yl]!j+1Yl=1(yi;1)=j+1Yl=1y�l:AformalstatementoftheresultandaproofrelyingonaconvergencedeterminingclassisgiveninthenextSection4.5.3.Table1givesasummaryoftheresultsintabularform. 28LINDSKOG,RESNICK,ANDROY(j)isgivenin(4.20),ormoreformally,(j)(A)=X(i1;:::;ij+1)ZInj+1Xk=1zkeik2Ao(dz1):::(dzj+1);wherethecomponentsofeikareallzeroexceptcomponentikwhosevalueis1andtheindices(i1;:::;ij+1)runthroughtheorderedsubsetsofsizej+1off1;2;:::g.TheproofofTheorem4.2usesaparticularconvergencedeterminingclassA�jofsubsetsofOj.LetA�jdenotethesetofsetsAm;i;aform�j,whereAm;i;a=fx2R1+:xik�akfork=1;:::;mg;i1


im;a1;:::;am&#x-278;0:Lemma4.1.Ift;2MOjandlimt!1t(A)=(A)forallA2A&#x-278;jboundedawayfromCjwith(@A)=0,thent!inMOjast!1.Proof.ConsiderthesetofnitedierencesofsetsinA&#x-278;jandnotethatthissetisa-system.Takex2Ojand&#x-278;0.Sincex2Ojtherearei1


ijsuchthatxik&#x-278;0foreachk.If2�ij=2choosem=ij.Otherwise,choosem&#x-348;ijsuchthat2�m=2.Takeminf=2;minfxk:xk&#x]TJ/;ø 9;&#x.962; Tf;&#x 18.;ˆ ;� Td;&#x [00;0andk6mggandsetB=fy2R1+:yk&#x]TJ/;ø 9;&#x.962; Tf;&#x 18.;ˆ ;� Td;&#x [00;0ifxk=0andyk&#x]TJ/;ø 9;&#x.962; Tf;&#x 18.;ˆ ;� Td;&#x [00;xk�otherwisefork6mgB0=fy2R1+:yk&#x]TJ/;ø 9;&#x.962; Tf;&#x 18.;ˆ ;� Td;&#x [00;ifxk=0andyk&#x]TJ/;ø 9;&#x.962; Tf;&#x 18.;ˆ ;� Td;&#x [00;xk+otherwisefork6mg:ThenB;B02A&#x]TJ/;ø 9;&#x.962; Tf;&#x 18.;ˆ ;� Td;&#x [00;j,B0isapropersubsetofB,andz2BrB0impliesthatd(z;x)Pmk=12�k+=2,i.e.thatz2Bx;.Moreover,(BrB0)=fy2R1+:yk2J(xk)fork6mg;whereJ(xk)=[0;)ifxk=0andJ(xk)=(xk�;xk+)ifxk6=0.Finally,@(BrB0)isthesetofy2R1+suchthatyk2[maxf0;xk�g;xk+]forallk6mandyk=oryk=xkforsomek6m.Inparticular,thereisanuncountablesetof-values,forwhichtheboundaries@(BrB0)aredisjoint,satisfyingtherequirements.Thereforecanwithoutlossofgeneralitybechosensothat(@(BrB0))=0.TheseparabilityofR1+implies(cf.theproofofTheorem2.3in[6])thateachopensetisacountableunionof-continuitysetsoftheform(BrB0).ThesameargumentasintheproofofTheorem2.2in[6]thereforeshowsthatliminft!1t(G)&#x-278;(G)forallopenGOjboundedawayfromCj.AnyclosedsetFOjboundedawayfromCjisasubsetofsomeA2A&#x-278;j.Bythesameargumentasabove,wemaywithoutlossofgeneralitytakeAsuchthat(@A)=0.ThesetArFisopenandtherefore(A)�limsupt!1t(F)=liminft!1t(ArF)&#x-278;(ArF)=(A)�(F);i.e.limsupt!1t(F)6(F).TheconclusionfollowsfromTheorem2.1(iii).ProofofTheorem4.2.Foranym&#x-278;janda1;:::;am&#x-278;0,limt!1c(t)jP(X2tAj;i;a)=jYk=1a�k=j(Aj;i;a)andlimt!1c(t)jP(tAj+1;i;a)=0: 30LINDSKOG,RESNICK,ANDROYForthersttwocomponents,letPRM()beaPoissoncountingfunctionwithmeanmeasureandforx�0;y�0,tP[Q (�1)=b(t)�x;Q (�2)=b(t)�y]6tP[PRM()(b(t)(x^y;1)�2]andwritingp(t)=(b(t)(x^y;1)),wehavetP[PRM()(b(t)(x^y;1)�2]=t(1�e�p(t)�p(t)e�p(t))6t(p(t)�p(t)e�p(t))6tp2(t)!0:TheconclusionnowfollowsfromLemma4.1byobservingthatwehaveshownconvergenceforthesetsinaconvergencedeterminingclass.Similarly,weclaimtP[�Q (�l)=b(t1=2);l�1
2
]!(2)(
)inM(O1)ast!1,where(2)(dx1dx2:::)=(dx1)(dx2)1[x1�x2�0]1Yl=30(dxl):Straightforwardcomputationsshowthatthedistributionof(�1;�2)=(E1;E1+E2)satisesP(�16z;�26w)=1�e�z�ze�w;zw;1�e�w�we�w;z&#x-278;w:Noticethat,forx&#x-278;y&#x-278;0,P[Q (�1)=b(t1=2)&#x-278;x;Q (�2)=b(t1=2)&#x-278;y]=P[�16Q(b(t1=2)x);�26Q(b(t1=2)y)]=1�e�Q(b(t1=2)x)�Q(b(t1=2)x)e�Q(b(t1=2)y)Q(b(t1=2)x)�Q(b(t1=2)x)2=2+O(Q(b(t1=2)x)3)�Q(b(t1=2)x)1�Q(b(t1=2)y)+O(Q(b(t1=2)y)2)Inparticular,itisastraightforwardexerciseincalculustoverifythatforx&#x-278;y&#x-278;0limt!1tP[Q (�1)=b(t1=2)&#x-278;x;Q (�2)=b(t1=2)&#x-278;y]=x�y��x�2=2=(2)(z2R1#:z1&#x-278;x;z2&#x-278;y):Similarcomputationsshowthat,fory&#x-278;x&#x-278;0,limt!1tP[Q (�1)=b(t1=2)&#x-278;x;Q (�2)=b(t1=2)&#x-278;y]=y�2=2=(2)(z2R1#:z1&#x-278;x;z2&#x-278;y):Moreover,forx&#x-278;0;y&#x-278;0;z&#x-278;0,tP[Q (�1)=b(t1=2)&#x-278;x;Q (�2)=b(t1=2)&#x-278;y;Q (�3)=b(t1=2)&#x-278;z] 32LINDSKOG,RESNICK,ANDROYThestandardItorepresentation[1,4,25]ofXisXt=ta+Bt+Zjxj61x[N([0;t]dx)�t(dx)]+Zjxj�1xN([0;t]dx);whereBisstandardBrownianmotionindependentofthePoissonrandommeasureNon[0;1](0;1)withmeanmeasureLeb.Referringtothediscussionpreceding(4.23),fQ (�n);n�1garepointswrittenindecreasingorderofaPoissonrandommeasureon(0;1)withmeanmeasureandbyaugmentation[39,p.122],wecanrepresentN=1Xl=1(Ul;Q (�l));where(Ul;l�1)areiidstandarduniformrandomvariablesindependentoff�ng.TheLevy-ItodecompositionallowsXtobedecomposedintothesumoftwoindependentLevyprocesses,(5.1)X=eX+J;whereJisacompoundPoissonprocessoflargejumpsboundedfrombelowby1,andeX=X�JisaLevyprocessofsmalljumpsthatareboundedfromaboveby1.ThecompoundPoissonprocesscanberepresentedastherandomsumJ=PN1l=1Q (�l)1[Ul;1],whereN1=N([0;1][1;1)).Recallthenotationin(4.23)forR1#+,H=jandH6jandtheresultinTheorem4.3.Weseektoconvertastatementlike(4.24)intoastatementaboutX.Therststepistoaugment(4.24)withasequenceofiidstandarduniformrandomvarables.TheuniformrandomvariableswilleventuallyserveasjumptimesfortheLevyprocess.ThefollowingresultisanimmediateconsequenceofTheorem4.3.Proposition5.1.UnderthegivenassumptionsonandQ,forj�1,tPh�(Q (�l)=b(t1=j);l�1);(Ul;l�1)
2
i!((j)L)(
)(5.2)inM((R1#+rH6j�1)[0;1]1)ast!1,whereLisLebesguemeasureon[0;1]1and(j)concentratesonH=jandisgivenby(4.25).Thinkof(5.2)asregularvariationontheproductspaceR1#+[0;1]1whenmultiplicationbyascalarisdenedas(;(x;y))7!(x;y).RecallistheParetomeasureon(0;1)satisfying(x;1)=x�,forx�0,andwedenotebyjproductmeasuregeneratedbywithjfactors.Form�0,letD6mbethesubspaceoftheSkorohodspaceDconsistingofnondecreasingstepfunctionswithatmostmjumpsanddeneAmasAm=f(x;u)2R1#+[0;1]1(5.3):ui2(0;1)for16i6m;ui6=ujfori6=j;16i;j6mg:LetTmbethemap(5.4)Tm:Am7!DdenedbyTm(x;u)=mXi=1xi1[ui;1]; 34LINDSKOG,RESNICK,ANDROY5.1.Details.WenowprovidemoredetailfortheproofofTheorem5.1.Inthedecomposition(5.1),theprocesseXrepresentssmalljumpsthatshouldnotaectasymptotics.WemakethisprecisewiththenextLemma.Lemma5.1.Forj�1,andany�0,limsupt!1tPhsups2[0;1]jeXsj�b(t1=j)i=0:Proof.WerelyonSkorohod'sinequalityforLevyprocesses[8],[37,Section7.3].Fora�0,Phsups2[0;1]jeXsj�2ai6(1�c)�1P[jeX1j�a];wherec=sups2[0;1]P[jeXsj�a].Thus,sinceeX1hasallmomentsnite,foranym�1,tPhsups2[0;1]jeXsj�b(t1=j)i6t(1�c(t))�1P[jeX1j�b(t1=j)=2]6t(1�c(t))�1EjeX1jm bm(t1=j)(=2)m:Forlargeenoughm,t=bm(t1=j)!0ast!1andc(t):=sups2[0;1]P[jeXs�b(t1=j)=2]6sups2[0;1]EjeXsjm bm(t1=j)(=2)m=sups2[0;1]smEjeX1jm bm(t1=j)(=2)m6EjeX1jm bm(t1=j)(=2)m!0;ast!1sinceb(t)!1.Lemma5.2.Forj�1,tP[J2b(t1=j)
]!((j)L)T�1j(
)inM(DrD6j�1)ast!1.Proof.WeapplyTheorem2.1(iii).Constructionofthelowerboundforopensets:LetGDbeopenandboundedawayfromD6j�1.ThisimpliesthatfunctionsinGhavenofewerthanjjumps.Recallthat�l=E1+


+El,wheretheEksareiidstandardexponentials.TakeM�jandnoticethattPhN1Xl=1Q (�l)1[Ul;1]2b(t1=j)Gi�tPhN1Xl=1Q (�l)1[Ul;1]2b(t1=j)G;N16Mi=tPhN1Xl=1Q (�l)1[Ul;1]2b(t1=j)G;j6N16Mi 36LINDSKOG,RESNICK,ANDROYDecomposetherstsummandaccordingtowhetherMt6jorMt�j+1.NoticeMtjisincompatiblewithPMtl=1Q (�l)1[Ul;1]2b(t1=j)FsinceFisboundedawayfromD6j�1.Thuswegettheupperbound6tPhjXl=1Q (�l)1[Ul;1]2b(t1=j)Fi+tP[Mt&#x-374;j+1]+tPhN1Xl=Mt+1Q (�l)&#x-374;b(t1=j)i:Wenowshowthatthesecondandthirdofthethreetermsabovevanishast!1.Firstly,thedenitionofMtimpliesthatQ (�l)6b(t1=j)forMt+16l6N1.Thus,tPhN1Xl=Mt+1Q (�l)&#x-374;b(t1=j)i6tP[(N1�Mt)b(t1=j)&#x-374;b(t1=j)]6tP[N1&#x-374;b(t1=j)1�]:Theright-handsideconvergesto0ast!1sincethetailprobabilityhasaMarkovboundoftE(N1)p=[b(t1=j)1�]pforanyp.Secondly,P[Mt&#x-374;j+1]6P[Q (�j+1)&#x-374;b(t1=j)]6P[�j+16([b(t1=j);1))]6P[max(E1;:::;Ej+1)6([b(t1=j);1))]=P[E16([b(t1=j);1))]j+1:SinceP[E16y]yasy#0,andsince([x;1))isregularlyvaryingatinnitywithindex�andbisregularlyvaryingatinnitywithindex1=,wendthatlimsupt!1t([b(t1=j);1))j+1=limsupt!1L(t)t1�(j+1)=jforsomeslowlyvaryingfunctionL.Inparticular,choosing2(j j+1;1)ensuresthatlimt!1tP[Mt�j+1]=0.Wenowdealwiththeremainingterm.SincetPhjXl=1Q (�l)1[Ul;1]2b(t1=j)Fi=tPh((Q (�l);l�1);(Ul;l�1))2b(t1=j)T�1j(F)i+tPh((Q (�l);l�1);(Ul;l�1))2Acmiand,byLemmas5.3and5.4,T�1j(F)is,ifnonempty,closedandboundedawayfromH6j�1[0;1]1,Proposition5.1andthefactthatAcmisaP-nullsetyield 38LINDSKOG,RESNICK,ANDROYProof.IfA\D6m=;,thenT�1m(A)=;.Therefore,withoutlossofgeneralitywemaytakeAD6m.Assumedsk(A;D6j�1)��0andnoticethatx2D6mifandonlyifx=mXi=1yi1[ui;1]fory1�:::�ym�0;ui2[0;1]:Ifx2A,Pmi=jyi�asaconsequenceofdsk(A;D6j�1)�andbecausethey'sarenon-increasing,yj�=(m�j+1).Consequently,T�1m(A)n�xi;i�1
2R1#+:xj�=(m�j+1)o[0;1]1;andthelattersetisboundedawayfromH6j�1[0;1]1.References[1]D.Applebaum.LevyProcessesandStochasticCalculus,volume93ofCam-bridgeStudiesinAdvancedMathematics.CambridgeUniversityPress,Cam-bridge,2004.ISBN0-521-83263-2.[2]A.A.Balkema.MonotoneTransformationsandLimitLaws.MathematischCentrum,Amsterdam,1973.MathematicalCentreTracts,No.45.[3]AABalkemaandP.Embrechts.Highriskscenariosandextremes:ageometricapproach.EuropeanMathematicalSociety,2007.[4]J.Bertoin.LevyProcesses,volume121ofCambridgeTractsinMathematics.CambridgeUniversityPress,Cambridge,1996.ISBN0-521-56243-0.[5]P.Billingsley.ProbabilityandMeasure.WileySeriesinProbabilityandMath-ematicalStatistics.JohnWiley&SonsInc.,NewYork,thirdedition,1995.ISBN0-471-00710-2.AWiley-IntersciencePublication.[6]P.Billingsley.ConvergenceofProbabilityMeasures.JohnWiley&SonsInc.,NewYork,secondedition,1999.ISBN0-471-19745-9.AWiley-IntersciencePublication.[7]N.H.Bingham,C.M.Goldie,andJ.L.Teugels.RegularVariation.CambridgeUniversityPress,1987.[8]L.Breiman.Probability,volume7ofClassicsinAppliedMathematics.SocietyforIndustrialandAppliedMathematics(SIAM),Philadelphia,PA,1992.ISBN0-89871-296-3.doi:10.1137/1.9781611971286.URLhttp://dx.doi.org/10.1137/1.9781611971286.Correctedreprintofthe1968original.[9]J.T.BruunandJ.A.Tawn.Comparisonofapproachesforestimatingtheprobabilityofcoastal ooding.J.R.Stat.Soc.,Ser.C,Appl.Stat.,47(3):405{423,1998.[10]S.G.Coles,J.E.Heernan,andJ.A.Tawn.Dependencemeasuresforextremevalueanalyses.Extremes,2(4):339{365,1999.[11]D.J.DaleyandD.Vere-Jones.AnIntroductiontotheTheoryofPointPro-cesses.SpringerSeriesinStatistics.Springer-Verlag,NewYork,1988.ISBN0-387-96666-8. 40LINDSKOG,RESNICK,ANDROY0-471-35629-8.[31]A.MitraandS.I.Resnick.HiddenRegularVariation:DetectionandEstima-tion.ArXive-prints,January2010.[32]A.MitraandS.I.Resnick.Hiddenregularvariationanddetectionofhiddenrisks.StochasticModels,27(4):591{614,2011.[33]A.MitraandS.I.Resnick.Modelingmultiplerisks:Hiddendomainofat-traction.Extremes,2013.doi:10.1007/s10687-013-0171-8.URLhttp:dx.doi.org/10.1007/s10687-013-0171-8.[34]L.Peng.Estimationofthecoecientoftaildependenceinbivariateextremes.Statist.Probab.Lett.,43(4):399{409,1999.ISSN0167-7152.[35]S.ResnickandD.Zeber.MarkovKernelsandtheConditionalExtremeValueModel.ArXive-prints,October2012.[36]S.I.Resnick.Pointprocesses,regularvariationandweakconvergence.Adv.AppliedProbability,18:66{138,1986.[37]S.I.Resnick.AProbabilityPath.Birkhauser,Boston,1999.[38]S.I.Resnick.Hiddenregularvariation,secondorderregularvariationandasymptoticindependence.Extremes,5(4):303{336(2003),2002.ISSN1386-1999.[39]S.I.Resnick.HeavyTailPhenomena:ProbabilisticandStatisticalModeling.SpringerSeriesinOperationsResearchandFinancialEngineering.Springer-Verlag,NewYork,2007.ISBN:0-387-24272-4.[40]S.I.Resnick.Multivariateregularvariationoncones:applicationtoextremevalues,hiddenregularvariationandconditionedlimitlaws.Stochastics:AnInternationalJournalofProbabilityandStochasticProcesses,80(2):269{298,2008.http://www.informaworld.com/10.1080/17442500701830423.[41]S.I.Resnick.ExtremeValues,RegularVariationandPointProcesses.Springer,NewYork,2008.ISBN978-0-387-75952-4.Reprintofthe1987original.[42]H.L.Royden.RealAnalysis.Macmillan,thirdedition,1988.[43]M.Schlather.Examplesforthecoecientoftaildependenceandthedomainofattractionofabivariateextremevaluedistribution.Stat.Probab.Lett.,53(3):325{329,2001.[44]E.Seneta.RegularlyVaryingFunctions.Springer-Verlag,NewYork,1976.LectureNotesinMathematics,508.[45]M.Sibuya.Bivariateextremestatistics.Ann.Inst.Stat.Math.,11:195{210,1960.FilipLindskog,DepartmentofMathematics,KTHRoyalInstituteofTechnology,10044Stockholm,SwedenE-mailaddress:lindskog@kth.seSidneyI.Resnick,SchoolofORIE,CornellUniversity,Ithaca,NY14853E-mailaddress:sir1@cornell.eduJoyjitRoy,SchoolofORIE,CornellUniversity,Ithaca,NY14853E-mailaddress:jr653@cornell.edu