Heavy tails of a Levy process and its maximum over a random time interval

SCIENCE CHINA Mathematics.ARTICLES .September 2011Vol.54No.9:1875–1884doi:10.1007/s11425-011-4223-8cScience China Press and Springer-Verlag Berlin Heidelberg Heavy tails of a L´e vy process and its maximum over a random time intervalLIU Yan 1,∗&TANG QiHe 21School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China;2Department of Statistics and Actuarial Science,The University of Iowa,Iowa City,IA 52242,USAEmail:yanliu@,qihe-tang@Received September 13,2010;accepted February 21,2011;published online August 1,2011Abstract Let {X t ,t 0}be a L´e vy process with L´e vy measure νon (−∞,∞),and let τbe a nonnegative random variable independent of {X t ,t 0}.We are interested in the tail probabilities of X τand X (τ)=sup 0 t τX t .For various cases,under the assumption that either the L´e vy measure νor the random variable τhas a heavy right tail we prove that both Pr(X τ>x )and Pr(X (τ)>x )are asymptotic to E τν((x,∞))+Pr(τ>x/(0∨E X 1))as x →∞,where Pr(τ>x/0)=0by convention.Keywordsheavy-tailed distribution,L´e vy process,maximum,tail asymptotics MSC(2000):60G51,60E07,60F10Citation:Liu Y,Tang Q H.Heavy tails of a L´e vy process and its maximum over a random time interval.Sci ChinaMath,2011,54(9):1875–1884,doi:10.1007/s11425-011-4223-81IntroductionL´e vy processes,as continuous-time analogues of random walks,are stochastic processes that start from 0,have stationary and independent increments,and are stochastically continuous.Textbook treatments of L´e vy processes are given in the monographs [5,14,24,32].Due to their important applications in various fields such as finance,insurance,physics,and engineering,L´e vy processes have been extensively studied by many researchers.We refer to [3,6,33]for comprehensive overviews of applications of L´e vy processes.For a L´e vy process X ={X t ,t 0},by definition,for each t 0the random variable X t has an infinitely divisible distribution with a characteristic function E(e i sX t )=e −t Φ(s ),where the characteristic exponent Φ(·)assumes the L´e vy-Khintchine representationΦ(s )=−i as +12bs 2+ ∞−∞(1−e i sx +i sx 1(|x | 1))ν(d x )(1.1)with a ∈(−∞,∞),b ∈[0,∞),and L´e vy measure νon (−∞,∞)satisfying ν({0})=0and 1−1x 2ν(d x )<∞.The triplet (a,b,ν)uniquely determines the law of the L´e vy process X .In the sequel,we write X (t )=sup 0 s t X s for t 0and ν(x )=ν((x,∞))for x >0.There has been a long history of the study of asymptotic tail behavior of L´e vy processes.The following result forms a core of this study:∗Corresponding author1876Liu Y et al.Sci China Math September2011Vol.54No.9Proposition1.1.Let the L´e vy process X have a L´e vy measureνwith a subexponential tailν(see below for this and other definitions).Then,for eachfixed t>0,Pr(X(t)>x)∼Pr(X t>x)∼tν(x).(1.2) Here and henceforth,all limit relationships are for x→∞unless stated otherwise and the symbol∼means that the quotient of both sides tends to1.A result similar to Proposition1.1but only for subordinators wasfirst discovered by[17].[38]proved that a long tail of the L´e vy measureνsuffices for thefirst relation of(1.2);see also[4],who obtained thefirst relation of(1.2)under stronger conditions.Proposition1.1in its present form is attributed to[31].This result has been extended to the case of light-tailed L´e vy measures by a series of papers; see[8,9,11,28,29],among others.From these references we see that,for the light-tailed case,it is usually possible to establish an asymptotic relation for Pr(X t>x)but hard to establish an asymptotic relation for Pr(X(t)>x).See also[1]for a recent overview of this study.We remark that all the above-mentioned references consider the tail probabilities of a L´e vy process at afixed time and its maximum over a compact time interval.Another direction of the mainstream study is to look at the tail behavior of some subadditive functionals of a L´e vy process;see[10,12,13,31],and references therein.In this paper,we aim to extend this study to the case of a random time.Let the L´e vy process X have afinite mean function E X t=mt for t 0,and letτbe a nonnegative random variable,called a random time,independent of X and with a distribution G which is not degenerate at0.Our goal is to derive asymptotic relations for the tail probabilities of Xτand X(τ)under the assumption that either the L´e vy measureνhas a heavy tailνor the random timeτhas a heavy tail G=1−G.Our main results belowshow that the relationsPr(X(τ)>x)∼Pr(Xτ>x)∼Eτν(x)+Gxm∨0(1.3)hold true for various cases,where G(x/0)=0by convention.Recently,[30]and[2]studied the tail probability of the random sum Sτ= τk=1X k,where{X1,X2,...}is a sequence of independent,identically distributed,and nonnegative random variables,andτis a nonnegative,integer-valued,and heavy-tailed random variable independent of the sequence{X1,X2,...}. They obtained some results similar to the second relation of(1.3).Our research is motivated by these works.We would like to point out that it is feasible to extend some known results to the case of both a light-tailed L´e vy measure and a random time.However,to keep the paper short we shall not pursue such an extension.The rest of this paper consists of four sections:Section2presents our main results after a brief review of heavy-tailed distributions,Section3prepares some lemmas,and Section4proves the main results.2Main results2.1A brief review of heavy-tailed distributionsA distribution F on[0,∞)or its corresponding random variable X is said to be heavy tailed if Ee rX=∞0−e rx F(d x)=∞for all r>0.A necessary condition for F to be heavy tailed is F(x)>0for all x 0.One of the most important classes of heavy-tailed distributions is the subexponential class,written as S,which is characterized by the relationlim x→∞F n∗(x)F(x)=n,for some(or,equivalently,for all)n=2,3,...,where F n∗denotes the n-fold convolution of F.It is well known that every subexponential distribution F is long tailed,written as F∈L,in the sense that theLiu Y et al.Sci China Math September2011Vol.54No.91877relationlimx→∞F(x−y)F(x)=1holds for some(or,equivalently,for all)y=0.Kl¨u ppelberg[22]introduced the class S∗,which ischaracterized bylimx→∞ xF(x−y)F(y)F(x)d y=2∞F(y)d y<∞,and she pointed out that the class S∗contains almost all cited subexponential distributions withfinite means.Closely related is the class S∗of so-called strongly subexponential distributions,introduced by[23]. By definition,a distribution F withfinite mean is said to belong to the class S∗if the relationlim x→∞F l∗F l(x)F l(x)=2holds uniformly for all l∈[1,∞],where F l(x)=1∧ x+lxF(y)d y.It is easy to see that S∗⊂S.Furthermore,Lemma9of[15]showed that S∗⊂S∗.Let D denote the class of distributions with dominatedly-varying tails in the sense that the relationlim x→∞F(xy)F(x)<∞holds for some(or,equivalently,for all)0<y<1.The intersection D∩L forms a useful subclass of S. Marginally smaller than D∩L is the class C of distributions with consistently-varying tails characterized bylim y↓1lim infx→∞F(xy)F(x)=1.Note that the class C contains all distributions with regularly-varying tails.Although it is customary that these classes are defined for distributions on[0,∞),in this paper we still say that a distribution F on(−∞,∞)(or,more generally,a measureνon(−∞,∞)with0<ν(x)<∞for all large x)belongs to one of these classes if the tail F(x)(respectively,ν(x))is asymptotic to the tail of a distribution on[0,∞)belonging to the same class.The reader is referred to[18]for an overview of heavy-tailed distributions with applications to insurance andfinance.2.2TheoremsThefirst main result below assumes a heavy-tailed L´e vy measure:Theorem2.1.Let the L´e vy process X have a L´e vy measureνand afinite mean function E X t=mt for t 0,and let the random timeτbe independent of X.The relationsPr(X(τ)>x)∼Pr(Xτ>x)∼Eτν(x)(2.1) hold under one of the following three sets of conditions:(a)ν∈S∗,m<0,and Eτ<∞;(b)ν∈L∩D,m=0,andlim y↑∞lim supx→∞G(xy)ν(x)=0;(2.2)(c)ν∈L∩D and G(x)=o(ν(x)).Note that thefiniteness of Eτis automatic in Theorem2.1(b)and(c);see Lemma3.3below.Recall that the lower Matuszewska index of1/νis defined to beJ−ν=inf−log˜ν(y)log y:y 1,where˜ν(y)=lim supx→∞ν(xy)ν(x).1878Liu Y et al.Sci China Math September 2011Vol.54No.9Assume 0<J −ν ∞,which is satisfied by almost all useful L´e vy measures with ultimate right tails.Then by Proposition 2.2.1of [7],for any 0<p <J −ν,there exist positive constants C and D such that the inequality ν(xy )ν(x )Cy −p holds for all xy x D ;see also the inequality (3.19)of [35].Hence,Relation (2.2)holds if we further assume G (x )=O (ν(x )).We point out that the results similar to (2.1)for a random walk with heavy-tailed increments and for a more general integer-valued random time τare well known in the literature.Some recent results in this aspect can be seen in [16,20,21,26,27],among others.The second main result below assumes both a heavy-tailed L´e vy measure and a heavy-tailed random time:Theorem 2.2.Let the L´e vy process X have a L´e vy measure νand a finite mean function E X t =mt for t 0,and let the random time τbe independent of X .If ν∈L ∩D ,m >0,E τ<∞,and G ∈C ,then Pr(X (τ)>x )∼Pr (X τ>x )∼E τν(x )+G x m.(2.3)Lemma 4.7of [19]shows a relation similar to the second one of (2.3)for a random walk with nonnegative increments under the assumptions that the tail of the common distribution of increments is regularly varying and is equivalent to that of an integer-valued random time.[2]further extended this result to the class C but still for a random walk with nonnegative increments.Theorem 7of [16],in the framework of a random walk with positive mean and real-valued,subexponential increments,is comparable to and stronger than our Theorem 2.2.The last main result below assumes a heavy-tailed random time:Theorem 2.3.Let the L´e vy process X have a L´e vy measure νand a finite mean function E X t =mt for t 0,and let the random time τbe independent of X .Assume G ∈C and E τ<∞.(a)If m >0and ν(x )=o (G (x ))then Pr(X (τ)>x )∼Pr (X τ>x )∼G x m;(2.4)(b)If m <0and ν(x )=o (G (x ))thenPr(X (τ)>x )=o (1)G (x );(2.5)(c)If m =0and ν((−∞,−x )∪(x,∞))=o (G (x ))then Pr sup 0 t τ|X t |>x ∼Pr(|X τ|>x )∼G x |m | .(2.6)Recently,[30]obtained a result corresponding to the second relation of (2.4)for a random walk under conditions similar to those of Theorem 2.3(a)but still assuming nonnegative increments.We end this section by concluding that the relations in (1.3)hold under the conditions of one of Theorems 2.1,2.2,and 2.3(a).3LemmasFor two positive functions f (·)and g (·),the relation f (x )∼g (x )amounts to the conjunction of the relations lim sup f (x )/g (x ) 1and lim inf f (x )/g (x ) 1,which are denoted by f (x ) g (x )and f (x ) g (x ),respectively.In the sequel,f (x ) g (x )means the conjunction of f (x )=O (g (x ))and g (x )=O (f (x )).Furthermore,for two positive bivariate functions f (·;·)and g (·;·),we say that the asymptotic relation f (x ;t )∼g (x ;t )holds uniformly over all t in a nonempty set Δif lim x →∞sup t ∈Δ f (x ;t )g (x ;t )−1 =0.Liu Y et al.Sci China Math September 2011Vol.54No.91879We define other uniform asymptotic relations such as f (x ;t ) g (x ;t )and f (x ;t ) g (x ;t )in a similar way.Lemma 3.1.Let F 1and F 2be two distributions on (−∞,∞)satisfying F 2(x )=O (F 1(x )).Then F 1∗F 2∈D if and only if F 1∈D ,and for each case we have F 1∗F 2(x ) F 1(x ).Proof.Observe thatF 1(3x/2)∼F 1(3x/2)F 2(−x/2) F 1∗F 2(x ) F 1(x/2)+F 2(x/2)=O (1)F 1(x/2).Hence,the claimed equivalence and the associated relation are proved.2Lemma 3.2.Let X be a L´e vy process.Then,for all t 0and x >x 0>0,Pr(X (t )>x )Pr inf 0 s t X s >−x 0 Pr(X t >x −x 0).Proof.See the Lemma of [38].2Recall the L´e vy process X introduced in Section 1.From (1.1),we can decompose X into two inde-pendent L´e vy processes asX t =Y t +Z t ,t 0,(3.1)where Y has a Laplace transform satisfying−log E(e −sY t )=t∞1(1−e −sx )ν(d x ),s,t 0,(3.2)and Z has a moment generating function satisfyinglog E(e sZ t )=t as +12bs 2− 1−∞(1−e sx +sx 1(|x | 1))ν(d x ) ,s,t 0.(3.3)See also [28].Then Y is a compound Poisson process with intensity ν(1)and jump distribution ν0(x )=ν((1,x ))/ν(1)for all x >1and 0otherwise.For each t 0,the random variable Z t has a light tail because E(e sZ t )<∞for all s 0.Lemma 3.3.Let the L´e vy process X be characterized by the triplet (a,b,ν)and recall the decomposition of (3.1).The following assertions are equivalent :(a)ν∈D ;(b)the distribution of Y 1belongs to D ;(c)the distribution of X 1belongs to D ;(d)for every t >0,the distribution of X t belongs to D .Furthermore,under one of these conditions,it holds for every fixed t >0thatν(x ) Pr (Y 1>x ) Pr (X 1>x ) Pr (X t >x ).Proof.The equivalence of (a)and (b)and the associated relation ν(x ) Pr (Y 1>x )are known from Proposition 4.1of [37];see also Theorem C of [34].The equivalence of (b)and (c)and the associated relation Pr (Y 1>x ) Pr (X 1>x )are immediate consequences of Lemma 3.1and the fact that Z 1is light tailed.It remains to prove the equivalence of (c)and (d)and the associated relation Pr (X 1>x ) Pr (X t >x )for every fixed t >0.Without loss of generality,we assume t ∈(0,1].By Lemma 3.2with x 0=x/2,Pr(X t >x ) Pr(X (1)>x ) Pr (X 1>x/2)Pr (inf 0 t 1X t >−x/2)∼Pr (X 1>x/2).(3.4)Similarly,for a temporarily fixed positive integer n 0such that 1/n 0 t ,Pr(X 1/n 0>x ) Pr(X (t )>x ) Pr X t >x 2.1880Liu Y et al.Sci China Math September2011Vol.54No.9 Hence,Pr(X t>x) Pr(X1/n0>2x)1n0Pr(X1>2n0x),(3.5)where the last step follows from the observationn0 j=1X j/n−X(j−1)/n>2x⊃(X1>2n0x).The combination of(3.4)and(3.5)completes the proof.2 Lemma3.4.Let the L´e vy process X have a L´e vy measureνand afinite mean function E X t=mt for t 0.Ifν∈D,then for eachfixedγ>m∨0there exist some constants C1,C2>0independent of x and t such that the inequalitiesPr(X t>x) C1(t+1)Pr(X1>x) C2(t+1)ν(x)(3.6)hold for all x γt and t 0.Proof.By Lemma3.3,the distribution of X1belongs to the class D and the relation Pr(X1>x) ν(x) holds.Thus,the second inequality of(3.6)trivially holds.We only prove thefirst inequality of(3.6).Choose someε>0such that(1−ε)γ>m∨0.For all x 0and t 0,we havePr(X t>x) Pr(X t >(1−ε)x)+Pr(X t− t >εx),(3.7) where t denotes the integer part of t.By Theorem1of[36],there exists a constant C independent of x and t such that the inequalityPr(X t >(1−ε)x) C t Pr(X1>(1−ε)x)(3.8) holds for all x γt and t 0.For the second term on the right-hand side of(3.7),by Lemma3.2,for arbitrarilyfixed x0>0and all x 2x0/ε>0,Pr(X t− t >εx) Pr(X(1)>εx)Pr(X1>εx/2)Pr(inf0 t 1X t>−x0).(3.9)Plugging(3.8)and(3.9)into(3.7)and using the fact that the tail Pr(X1>x)is dominatedly varying, we obtain that,for some constant C1independent of x and t,thefirst inequality of(3.6)holds for all x (γt)∨(2x0/ε)and t 0.The extension to the whole area x γt and t 0is straightforward.2 4Proofs4.1Proof of Theorem2.1Note that,by Proposition1.1and Fatou’s lemma,Pr(X(τ)>x) Pr(Xτ>x)=∞Pr(X t>x)G(d t) Eτν(x).To complete the proof of(2.1),it suffices to show thatPr(X(τ)>x)∼Eτν(x).(4.1)(a)By Proposition1.1,the distribution of X1belongs to the class S∗.Following the same lines of the proof of Proposition4.3of[25]with some obvious adjustments,we obtain that,for all n=1,2,...andall x>x0>0,Pr(X(n)>x)P0 Prsupk=0,1,...,nX k>x−x0,Liu Y et al.Sci China Math September 2011Vol.54No.91881where P 0=Pr (inf 0 s<1X s >−x 0).By this and the theorem of [23],it holds uniformly for all t 0thatPr(X (t )>x ) Pr(X ( t +1)>x ) 1P 0Pr sup k =0,1,..., t +1X k >x −x 0 ∼1P 01|m | x +( t +1)|m |x Pr (X 1>y −x 0)d y 1P 0(t +1)Pr (X 1>x ).This,together with Proposition 1.1,justifies the validity of applying the dominated convergence theorem in deriving lim x →∞Pr(X (τ)>x )ν(x )= ∞0lim x →∞Pr(X (t )>x )ν(x )G (d t )=E τ,yielding relation (4.1).(b)For arbitrarily fixed ε>0and 0<y <(2ε)−1,write Pr(X (τ)>x )= xy 0+∞xyPr(X (t )>x )G (d t ).(4.2)Choose 1/2<θ<1so that θ−εy >0.From Lemma 3.2with x 0=θx ,uniformly for all 0 t xy ,Pr(X (t )>x ) Pr (X t >(1−θ)x )Pr (inf 0 s t X s >−θx ) Pr (X t >(1−θ)x )Pr (inf 0 s t (X s +εs )>−θx +εt )Pr (X t >(1−θ)x )Pr (inf 0 s<∞(X s +εs )>−(θ−εy )x )∼Pr (X t >(1−θ)x ).By this and Lemma 3.4,there exists some constant C independent of x and t such that the relationPr(X (t )>x ) C (t +1)ν(x )(4.3)holds uniformly for all 0 t xy .Applying the dominated convergence theorem and Proposition 1.1,we obtain that xyPr(X (t )>x )G (d t )∼E τν(x ).(4.4)By Condition (2.2),lim y ↑∞lim sup x →∞ ∞xy Pr(X (t )>x )G (d t )ν(x )=0.(4.5)Substituting (4.4)and (4.5)into (4.2)yields relation (4.1).(c)Since the case m =0has been dealt with in Theorem 2.1(b),now we assume m =0.Continue to employ (4.2)in which y is a positive constant to be determined later.Since,for arbitrarily fixed y >0, ∞xyPr(X (t )>x )G (d t ) G (xy )=o (ν(xy ))=o (ν(x )),it suffices to prove that relation (4.4)holds for suitably chosen y >0.When m >0,choose 0<y <(2m )−1.By Lemma 3.2with x 0=x/2,it holds uniformly for all t >0thatPr(X (t )>x ) Pr (X t >x/2).By this and Lemma 3.4,relation (4.3)holds uniformly for all 0 t xy .Applying the dominated convergence theorem and Proposition 1.1,we obtain relation (4.4).When m <0,for arbitrarily fixed ε>0we choose y and θsuch that 0<(1+ε)|m |y <θ<1.Note that,uniformly for all 0 t xy ,Pr inf 0 s t X s >−θx Pr inf 0 s t (X s +(1+ε)|m |s )>−θx +(1+ε)|m |t Pr inf 0 s<∞(X s +(1+ε)|m |s )>−(θ−(1+ε)|m |y )x →1.By Lemma 3.2with x 0=θx and Lemma 3.4,Relation (4.3)holds uniformly for all 0 t xy .Therefore,applying the dominated convergence theorem and Proposition 1.1,we obtain relation (4.4)again.1882Liu Y et al.Sci China Math September 2011Vol.54No.94.2Proof of Theorem 2.2We prove,in turn,the relations Pr(X (τ)>x ) E τν(x )+G x mPr (X τ>x ).(4.6)For arbitrarily fixed 0<ε<m ,write Pr(X (τ)>x )= x m +ε0+ ∞x m +εPr(X (t )>x )G (d t )=I 1(x )+I 2(x ).For the probability in the integral of I 1(x ),by Lemma 3.2with x 0=ε(2(m +ε))−1x and Lemma 3.4,there exists some constant C independent of x and t such that the relationsPr(X (t )>x ) Pr X t > 1−ε2(m +ε) x C (t +1)ν(x )hold uniformly for all 0<t x/(m +ε).Applying the dominated convergence theorem and Proposi-tion 1.1,we obtain I 1(x )∼E τν(x ).Since G ∈C ,it follows thatlim sup ε↓0lim sup x →∞I 2(x )G (x/m ) lim sup ε↓0lim sup x →∞G (x/(m +ε))G (x/m )=1.This proves the first relation in (4.6).We turn to the second relation of (4.6).Similarly,for arbitrarily fixed 0<ε<m ,writePr(X τ>x )=x m −ε0+ ∞x m −ε Pr (X t >x )G (d t )=I 3(x )+I 4(x ).By Proposition 1.1and Fatou’s lemma,I 3(x ) E τν(x ).For the term I 4(x ),we deriveI 4(x )= ∞x m −εPr X t t >x t G (d t )∼G x m −ε ,since,by the law of large numbers,the probability Pr (X t /t >x/t )in the integral tends to 1uniformly for all t >x/(m −ε).Consequently,lim inf ε↓0lim inf x →∞I 4(x )G (x/m )=lim inf ε↓0lim inf x →∞G (x/(m −ε))G (x/m )=1.This proves the second relation in (4.6).4.3Proof of Theorem 2.3Let the L´e vy process X be characterized by the triplet (a,b,ν).(a)For arbitrarily fixed 0<δ<1,introduce a new measure ν∗on (−∞,∞)such thatν∗(d x )=ν(d x ),x 1,ν∗(x )=ν(x )∨(δG (x )),x >1.Then ν∗is still a L´e vy measure since ν∗({0})=0and ∞−∞(x 2∧1)ν∗(d x )<∞.Let X ∗={X ∗t ,t 0}be a L´e vy process with triplet (a,b,ν∗)independent of τ.As done in (3.1),we decompose X ∗into twoindependent L´e vy processes as X ∗t =Y ∗t +Z ∗t ,t 0,where Y ∗has a Laplace transform satisfying (3.2)with νreplaced by ν∗and Z ∗has a moment generating function satisfying (3.3).Note the following facts:(1)ν∗∈C because ν∗(x )∼δG (x );Liu Y et al.Sci China Math September 2011Vol.54No.91883(2)X st X ∗,meaning that X is stochastically not larger than X ∗in terms of their finite dimensional distributions;(3)it holds for arbitrarily fixed 0<ε<|m |and for all sufficiently small δ>0thatm m ∗=E X ∗1 m +ε.(4.7)By Theorem 2.2,we havePr(X ∗(τ)>x )∼Pr (X ∗τ>x )∼E τν∗(x )+G x m ∗ ∼δE τG (x )+G x m ∗.Hence,there exists some constant C independent of x such that Pr(X (τ)>x ) Pr(X ∗(τ)>x ) (CδE τ+1)Gx m +ε .Since G ∈C and ε,δcan be arbitrarily close to 0,it follows that Pr(X (τ)>x ) G (x m ).For the corresponding lower bound,we derivePr (X τ>x ) ∞x m −εPr X t t >x t G (d t )∼G x m −ε ,where in the last step we have used the fact that the probability Pr (X t /t >x/t )in the integral tends to 1uniformly for all t >x/(m −ε).Since G ∈C and εcan be arbitrarily close to 0,it follows thatPr(X τ>x ) G (x m ).(b)Copy the proof of (a)until (4.7).Note that in the current case E X ∗1 m +ε<0,and ν∗∈S ∗⊂S ∗because G ∈S ∗by Theorem 3.2of [22].Therefore,by Theorem 2.1(a),Pr(X ∗(τ)>x )∼Pr (X ∗τ>x )∼E τν∗(x )∼δE τG (x ).Since δ>0can be arbitrarily small,it follows that Pr(X (τ)>x ) Pr(X ∗(τ)>x )=o (1)G (x ).(c)Without loss of generality,we assume m >bining (2.4)with (2.5)yields thatPr (|X τ|>x )=Pr (X τ>x )+Pr (−X τ>x )∼G x m.Similarly,Prsup 0 t τ|X t |>x Pr(X (τ)>x )+Pr sup 0 t τ(−X t )>x ∼G x m .This proves the relations in (2.6).Acknowledgements The authors would like to thank two anonymous referees for their thoughtful comments on an earlier version of this paper.The majority of this work was completed during a research visit of the first author to the University of Iowa.She would like to thank the Department of Statistics and Actuarial Science for its excellent hospitality.This work was partially supported by National Natural Science Foundation of China (Grant Nos.10971157,11001209,70871104)and the Scientific Research Foundation for the Returned Overseas Chinese Scholars.References1Albin J M P,Sund´e n M.On the asymptotic behaviour of L´e vy processes.I.Subexponential and exponential processes.Stochastic Process Appl,2009,119:281–3042Aleˇs keviˇc ien˙e A,Leipus R,ˇSiaulys J.Tail behavior of random sums under consistent variation with applications to thecompound renewal risk model.Extremes,2008,11:261–2793Barndorff-Nielsen O E,Mikosch T,Resnick S I.L´e vy 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