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

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1 SCIENCE CHINA Mathematics. ARTICLES. September 211 Vol. 54 No. 9: doi: 1.17/s Heavy tails of a Lévy process and its maimum over a random time interval LIU Yan 1, & TANG QiHe 2 1 School of Mathematics and Statistics, Wuhan University, Wuhan 4372, China; 2 Department of Statistics and Actuarial Science, The University of Iowa, Iowa City, IA 52242, USA yanliu@whu.edu.cn, qihe-tang@uiowa.edu Received September 13, 21; accepted February 21, 211; published online August 1, 211 Abstract Let {X t,t } be a Lévy process with Lévy measure ν on,, and let τ be a nonnegative random variable independent of {X t,t }. We are interested in the tail probabilities of X τ and X τ = sup t τ X t. For various cases, under the assumption that either the Lévy measure ν or the random variable τ has a heavy right tail we prove that both PrX τ >andprx τ > are asymptotic to Eτν, +Prτ > / EX 1 as,whereprτ>/ = by convention. Keywords MSC2: heavy-tailed distribution, Lévy process, maimum, tail asymptotics 6G51, 6E7, 6F1 Citation: Liu Y, Tang Q H. Heavy tails of a Lévy process and its maimum over a random time interval. Sci China Math, 211, 549: , doi: 1.17/s Introduction Lévy processes, as continuous-time analogues of random walks, are stochastic processes that start from, have stationary and independent increments, and are stochastically continuous. Tetbook treatments of Lévy 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, Lévy processes have been etensively studied by many researchers. We refer to [3,6,33] for comprehensive overviews of applications of Lévy processes. For a Lévy process X = {X t,t }, by definition, for each t the random variable X t has an infinitely divisible distribution with a characteristic function Ee isxt =e tφs, where the characteristic eponent Φ assumes the Lévy-Khintchine representation Φs = ias bs2 + 1 e is +is1 1 νd 1.1 with a,, b [,, and Lévy measure ν on, satisfying ν {} =and νd <. The triplet a, b, ν uniquely determines the law of the Lévy process X. In the sequel, we write X t =sup s t X s for t andν =ν, for >. There has been a long history of the study of asymptotic tail behavior of Lévy processes. The following result forms a core of this study: Corresponding author c Science China Press and Springer-Verlag Berlin Heidelberg 211 math.scichina.com

2 1876 Liu Y et al. Sci China Math September 211 Vol. 54 No. 9 Proposition 1.1. Let the Lévy process X have a Lévy measure ν with a subeponential tail ν see below for this and other definitions. Then, for each fied t>, PrX t > PrX t > tν. 1.2 Here and henceforth, all it relationships are for unless stated otherwise and the symbol means that the quotient of both sides tends to 1. A result similar to Proposition 1.1 but only for subordinators was first discovered by [17]. [38] proved that a long tail of the Lévy measure ν suffices for the first relation of 1.2; see also [4], who obtained the first relation of 1.2 under stronger conditions. Proposition 1.1 in its present form is attributed to [31]. This result has been etended to the case of light-tailed Lévy 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 PrX t > but hard to establish an asymptotic relation for PrX t >. See also [1] for a recent overview of this study. We remark that all the above-mentioned references consider the tail probabilities of a Lévy process at a fied time and its maimum over a compact time interval. Another direction of the mainstream study is to look at the tail behavior of some subadditive functionals of a Lévy process; see [1, 12, 13, 31], and references therein. In this paper, we aim to etend this study to the case of a random time. Let the Lévy process X have a finite mean function EX t = mt for t, and let τ be a nonnegative random variable, called a random time, independent of X and with a distribution G which is not degenerate at. Our goal is to derive asymptotic relations for the tail probabilities of X τ and X τ under the assumption that either the Lévy measure ν has a heavy tail ν or the random time τ has a heavy tail G =1 G. Our main results below show that the relations PrX τ > Pr X τ > Eτν+G 1.3 m hold true for various cases, where G / = by convention. Recently, [3] and [2] studied the tail probability of the random sum S τ = τ k=1 X k, where {X 1,X 2,...} is a sequence of independent, identically distributed, and nonnegative random variables, and τ is a nonnegative, integer-valued, and heavy-tailed random variable independent of the sequence {X 1,X 2,...}. 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 etend some known results to the case of both a light-tailed Lévy measure and a random time. However, to keep the paper short we shall not pursue such an etension. The rest of this paper consists of four sections: Section 2 presents our main results after a brief review of heavy-tailed distributions, Section 3 prepares some lemmas, and Section 4 proves the main results. 2 Main results 2.1 A brief review of heavy-tailed distributions A distribution F on [, or its corresponding random variable X is said to be heavy tailed if Ee rx = er F d = for all r>. A necessary condition for F to be heavy tailed is F > for all. One of the most important classes of heavy-tailed distributions is the subeponential class, written as S, which is characterized by the relation F n = n, F for some or, equivalently, for all n =2, 3,..., where F n denotes the n-fold convolution of F.Itiswell known that every subeponential distribution F is long tailed, written as F L, in the sense that the

3 Liu Y et al. Sci China Math September 211 Vol. 54 No relation F y =1 F holds for some or, equivalently, for all y. Klüppelberg [22] introduced the class S, which is characterized by F yf y dy =2 F ydy <, F and she pointed out that the class S contains almost all cited subeponential distributions with finite means. Closely related is the class S of so-called strongly subeponential distributions, introduced by [23]. By definition, a distribution F with finite mean is said to belong to the class S if the relation F l F l =2 F l holds uniformly for all l [1, ], where F l = 1 +l F ydy. It is easy to see that S S. Furthermore, Lemma 9 of [15] showed that S S. Let D denote the class of distributions with dominatedly-varying tails in the sense that the relation F y F < holds for some or, equivalently, for all <y<1. The intersection D Lforms a useful subclass of S. Marginally smaller than D L is the class C of distributions with consistently-varying tails characterized by F y inf y 1 F =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 [,, in this paper we still say that a distribution F on, or, more generally, a measure ν on, with< ν < for all large belongs to one of these classes if the tail F respectively, ν is asymptotic to the tail of a distribution on [, belonging to the same class. The reader is referred to [18] for an overview of heavy-tailed distributions with applications to insurance and finance. 2.2 Theorems Thefirstmainresultbelowassumesaheavy-tailedLévy measure: Theorem 2.1. Let the Lévy process X have a Lévy measure ν and a finite mean function EX t = mt for t, and let the random time τ be independent of X. The relations PrX τ > Pr X τ > Eτν 2.1 hold under one of the following three sets of conditions: a ν S, m<, andeτ < ; b ν L D, m =,and Gy sup = ; 2.2 y ν c ν L Dand G =oν. Note that the finiteness of Eτ is automatic in Theorem 2.1b and c; see Lemma 3.3 below. Recall that the lower Matuszewska inde of 1/ν is defined to be J ν =inf { log νy : y 1 log y }, where νy = sup νy ν.

4 1878 Liu Y et al. Sci China Math September 211 Vol. 54 No. 9 Assume <Jν, which is satisfied by almost all useful Lévy measures with ultimate right tails. Then by Proposition of [7], for any <p<jν, there eist positive constants C and D such that the inequality ν y ν Cy p holds for all y D; see also the inequality 3.19 of [35]. Hence, Relation 2.2 holds if we further assume G =Oν. 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 τ are well known in the literature. Some recent results in this aspect can be seen in [16, 2, 21, 26, 27], among others. The second main result below assumes both a heavy-tailed Lévy measure and a heavy-tailed random time: Theorem 2.2. Let the Lévy process X have a Lévy measure ν and a finite mean function EX t = mt for t, and let the random time τ be independent of X. If ν L D, m>, Eτ <, andg C, then PrX τ > Pr X τ > Eτν+G m. 2.3 Lemma 4.7 of [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 etended this result to the class C but still for a random walk with nonnegative increments. Theorem 7 of [16], in the framework of a random walk with positive mean and real-valued, subeponential 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 Lévy process X have a Lévy measure ν and a finite mean function EX t = mt for t, and let the random time τ be independent of X. Assume G C and Eτ <. a If m> and ν =og then PrX τ > Pr X τ > G ; 2.4 m b If m< and ν =og then PrX τ >=o1g; 2.5 c If m and ν,, = og then Pr sup X t > Pr X τ > G. 2.6 t τ m Recently, [3] obtained a result corresponding to the second relation of 2.4 for a random walk under conditions similar to those of Theorem 2.3a 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 2.3a. 3 Lemmas For two positive functions f andg, the relation f g amounts to the conjunction of the relations supf/g 1 and inff/g 1, which are denoted by f g andf g, respectively. In the sequel, f g means the conjunction of f =Og and g = Of. Furthermore, for two positive bivariate functions f ; andg ;, we say that the asymptotic relation f; t g; t holds uniformly over all t in a nonempty set Δ if sup f; t g; t 1 =. t Δ

5 Liu Y et al. Sci China Math September 211 Vol. 54 No We define other uniform asymptotic relations such as f; t g; t andf; t g; t in a similar way. Lemma 3.1. Let F 1 and F 2 be two distributions on, satisfying F 2 =OF 1. Then F 1 F 2 Dif and only if F 1 D, and for each case we have F 1 F 2 F 1. Proof. Observe that F 1 3/2 F 1 3/2 F 2 /2 F 1 F 2 F 1 /2 + F 2 /2 = O1F 1 /2. Hence, the claimed equivalence and the associated relation are proved. Lemma 3.2. Let X be a Lévy process. Then, for all t and > >, PrX t >Pr inf X s > PrX t >. s t Proof. See the Lemma of [38]. Recall the Lévy process X introduced in Section 1. From 1.1, we can decompose X into two independent Lévy processes as X t = Y t + Z t, t, 3.1 where Y has a Laplace transform satisfying log Ee syt =t 1 1 e s νd, s,t, 3.2 and Z has a moment generating function satisfying log Ee szt =t as bs2 1 e s + s1 1 νd, s,t. 3.3 See also [28]. Then Y is a compound Poisson process with intensity ν1 and jump distribution ν = ν1,/ν1 for all >1andotherwise. Foreacht, the random variable Z t has a light tail because Ee szt < for all s. Lemma 3.3. Let the Lévy 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 1 belongs to D; c the distribution of X 1 belongs to D; d for every t>, the distribution of X t belongs to D. Furthermore, under one of these conditions, it holds for every fied t> that ν Pr Y 1 > Pr X 1 > Pr X t >. Proof. The equivalence of a and b and the associated relation ν Pr Y 1 > are known from Proposition 4.1 of [37]; see also Theorem C of [34]. The equivalence of b and c and the associated relation Pr Y 1 > Pr X 1 > are immediate consequences of Lemma 3.1 and the fact that Z 1 is light tailed. It remains to prove the equivalence of c and d and the associated relation Pr X 1 > Pr X t > for every fied t>. Without loss of generality, we assume t, 1]. By Lemma 3.2 with = /2, PrX t > PrX 1 > Pr X 1 >/2 Pr inf t 1 X t > /2 Pr X 1 >/ Similarly, for a temporarily fied positive integer n such that 1/n t, PrX 1/n > PrX t > Pr X t >. 2

6 188 Liu Y et al. Sci China Math September 211 Vol. 54 No. 9 Hence, Pr X t > PrX 1/n > 2 1 n Pr X 1 > 2n, 3.5 where the last step follows from the observation n j=1 Xj/n X j 1/n > 2 X 1 > 2n. The combination of 3.4 and 3.5 completes the proof. Lemma 3.4. Let the Lévy process X have a Lévy measure ν and a finite mean function EX t = mt for t. If ν D, then for each fied γ>m there eist some constants C 1,C 2 > independent of and t such that the inequalities hold for all γt and t. Pr X t > C 1 t +1PrX 1 > C 2 t +1ν 3.6 Proof. By Lemma 3.3, the distribution of X 1 belongs to the class D and the relation Pr X 1 > ν holds. Thus, the second inequality of 3.6 trivially holds. We only prove the first inequality of 3.6. Choose some ε> such that 1 εγ >m. For all andt, we have PrX t > PrX t > 1 ε+prx t t >ε, 3.7 where t denotes the integer part of t. By Theorem 1 of [36], there eists a constant C independent of and t such that the inequality PrX t > 1 ε C t PrX 1 > 1 ε 3.8 holds for all γt and t. For the second term on the right-hand side of 3.7, by Lemma 3.2, for arbitrarily fied > andall 2 /ε >, PrX t t >ε PrX 1 >ε PrX 1 >ε/2 Prinf t 1 X t >. 3.9 Plugging 3.8 and 3.9 into 3.7 and using the fact that the tail Pr X 1 > is dominatedly varying, we obtain that, for some constant C 1 independent of and t, the first inequality of 3.6 holds for all γt 2 /ε andt. The etension to the whole area γt and t is straightforward. 4 Proofs 4.1 Proof of Theorem 2.1 Note that, by Proposition 1.1 and Fatou s lemma, PrX τ > Pr X τ >= To complete the proof of 2.1, it suffices to show that Pr X t > G dt Eτν. PrX τ > Eτν. 4.1 a By Proposition 1.1, the distribution of X 1 belongs to the class S. Following the same lines of the proof of Proposition 4.3 of [25] with some obvious adjustments, we obtain that, for all n =1, 2,... and all > >, PrX n >P Pr sup k=,1,...,n X k >,

7 Liu Y et al. Sci China Math September 211 Vol. 54 No where P = Pr inf s<1 X s >. By this and the theorem of [23], it holds uniformly for all t that PrX t > PrX t +1 > 1 Pr X k > P 1 P 1 m + t +1 m sup k=,1,..., t +1 Pr X 1 >y dy 1 P t +1PrX 1 >. This, together with Proposition 1.1, justifies the validity of applying the dominated convergence theorem in deriving PrX τ > PrX t > = G dt =Eτ, ν ν yielding relation 4.1. b For arbitrarily fied ε>and<y<2ε 1,write y PrX τ >= + PrX t >G dt. 4.2 Choose 1/2 <θ<1sothatθ εy >. From Lemma 3.2 with = θ, uniformly for all t y, PrX t > y Pr X t > 1 θ Pr inf s t X s > θ Pr X t > 1 θ Pr inf s t X s + εs > θ + εt Pr X t > 1 θ Pr inf s< X s + εs > θ εy Pr X t > 1 θ. By this and Lemma 3.4, there eists some constant C independent of and t such that the relation PrX t > C t +1ν 4.3 holds uniformly for all t y. Applying the dominated convergence theorem and Proposition 1.1, we obtain that y PrX t >G dt Eτν. 4.4 By Condition 2.2, y sup PrX t >Gdt =. 4.5 y ν Substituting 4.4 and 4.5 into 4.2 yields relation 4.1. c Since the case m = has been dealt with in Theorem 2.1b, now we assume m. Continueto employ 4.2 in which y is a positive constant to be determined later. Since, for arbitrarily fied y>, y PrX t >G dt Gy =o νy = o ν, it suffices to prove that relation 4.4 holds for suitably chosen y>. When m>, choose <y<2m 1. By Lemma 3.2 with = /2, it holds uniformly for all t> that PrX t > Pr X t >/2. By this and Lemma 3.4, relation 4.3 holds uniformly for all t y. Applying the dominated convergence theorem and Proposition 1.1, we obtain relation 4.4. When m<, for arbitrarily fied ε>wechoosey and θ such that < 1 + ε m y<θ<1. Note that, uniformly for all t y, Pr inf X s > θ Pr inf X s +1+ε m s > θ +1+ε m t s t s t Pr inf X s +1+ε m s > θ 1 + ε m y 1. s< By Lemma 3.2 with = θ and Lemma 3.4, Relation 4.3 holds uniformly for all t y. Therefore, applying the dominated convergence theorem and Proposition 1.1, we obtain relation 4.4 again.

8 1882 Liu Y et al. Sci China Math September 211 Vol. 54 No Proof of Theorem 2.2 We prove, in turn, the relations PrX τ > Eτν+G Pr X τ >. 4.6 m For arbitrarily fied <ε<m,write PrX τ >= m+ε + PrX t >G dt =I 1 +I 2. m+ε For the probability in the integral of I 1, by Lemma 3.2 with = ε 2m + ε 1 and Lemma 3.4, there eists some constant C independent of and t such that the relations ε PrX t > Pr X t > 1 C t +1ν 2m + ε hold uniformly for all <t /m + ε. Applying the dominated convergence theorem and Proposition 1.1, we obtain I 1 Eτν. Since G C, it follows that sup sup ε I 2 G /m G /m + ε sup sup =1. ε G /m This proves the first relation in 4.6. We turn to the second relation of 4.6. Similarly, for arbitrarily fied <ε<m,write m ε PrX τ >= + Pr X t > G dt =I 3 +I 4. m ε By Proposition 1.1 and Fatou s lemma, I 3 Eτν. For the term I 4, we derive I 4 = m ε Xt Pr > G dt G, t t m ε since, by the law of large numbers, the probability Pr X t /t > /t in the integral tends to 1 uniformly for all t>/m ε. Consequently, inf inf ε This proves the second relation in Proof of Theorem 2.3 I 4 G /m G /m ε = inf inf =1. ε G /m Let the Lévy process X be characterized by the triplet a, b, ν. a For arbitrarily fied <δ<1, introduce a new measure ν on, such that { ν d =νd, 1, ν =ν δg, > 1. Then ν is still a Lévy measure since ν {} =and 2 1ν d <. Let X = {Xt,t } be a Lévy process with triplet a, b, ν independent of τ. As done in 3.1, we decompose X into two independent Lévy processes as Xt = Y t + Z t, t, where Y has a Laplace transform satisfying 3.2 with ν replaced by ν and Z has a moment generating function satisfying 3.3. Note the following facts: 1 ν C because ν δg;

9 Liu Y et al. Sci China Math September 211 Vol. 54 No X st X, meaning that X is stochastically not larger than X in terms of their finite dimensional distributions; 3 it holds for arbitrarily fied <ε< m and for all sufficiently small δ>that m m =EX 1 m + ε. 4.7 By Theorem 2.2, we have PrXτ > Pr X τ > Eτν +G m δeτg+g m. Hence, there eists some constant C independent of such that PrX τ > PrXτ > CδEτ +1G. m + ε Since G Cand ε, δ can be arbitrarily close to, it follows that PrX τ > G m. For the corresponding lower bound, we derive Xt Pr X τ > Pr > G dt G, t t m ε m ε where in the last step we have used the fact that the probability Pr X t /t > /t in the integral tends to 1 uniformly for all t>/m ε. Since G Cand ε can be arbitrarily close to, it follows that PrX τ > G m. b Copy the proof of a until 4.7. Note that in the current case EX 1 m+ε <, and ν S S because G S by Theorem 3.2 of [22]. Therefore, by Theorem 2.1a, PrX τ > Pr X τ > Eτν δeτg. Since δ> can be arbitrarily small, it follows that PrX τ > PrX τ >= o1g. c Without loss of generality, we assume m>. Combining 2.4 with 2.5 yields that Pr X τ >=PrX τ >+Pr X τ > G. m Similarly, Pr sup t τ This proves the relations in 2.6. X t > PrX τ >+Pr sup t τ X t > G. m 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 ecellent hospitality. This work was partially supported by National Natural Science Foundation of China Grant Nos , 11129, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars. References 1 Albin J M P, Sundén M. On the asymptotic behaviour of Lévy processes. I. Subeponential and eponential processes. Stochastic Process Appl, 29, 119: Aleškevičienė A, Leipus R, Šiaulys J. Tail behavior of random sums under consistent variation with applications to the compound renewal risk model. Etremes, 28, 11: Barndorff-Nielsen O E, Mikosch T, Resnick S I. Lévy Processes. Boston: Birkhäuser Boston, Inc, 21 4 Berman S M. The supremum of a process with stationary independent and symmetric increments. Stochastic Process Appl, 1986, 23: Bertoin J. Lévy Processes. Cambridge: Cambridge University Press, 1996

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