Subexponential Tails of Discounted Aggregate Claims in a Time-Dependent Renewal Risk Model

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1 Subexponential Tails of Discounted Aggregate Claims in a Time-Dependent Renewal Risk Model Jinzhu Li [a];[b], Qihe Tang [b];, and Rong Wu [a] [a] School of Mathematical Science and LPMC Nankai University, Tianjin 37, P.R. China s: lijinzhunk@gmail.com (J. Li); rongwu@nankai.edu.cn (R. Wu) [b] Department of Statistics and Actuarial Science, The University of Iowa 24 Schae er Hall, Iowa City, IA 52242, USA qtang@stat.uiowa.edu August 23, 2 Abstract Consider a continuous-time renewal risk model with a constant force of interest. We assume that claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys a dependence structure described via the conditional tail probability of a claim size given the inter-arrival time before the claim. We focus on determination of the impact of this dependence structure on the asymptotic tail probability of discounted aggregate claims. Assuming that the claim-size distribution is subexponential, we derive an exact locally uniform asymptotic formula, which quantitatively captures the impact of the dependence structure. When the claim-size distribution is extended-regularly-varying tailed, we show that this asymptotic formula is globally uniform. Keywords: Asymptotics; Dependence; Discounted aggregate claims; Extendedregular variation; Subexponentiality; Uniformity. Mathematics Subject Classi cation: Primary 62P5; Secondary 62H2, 6E5 Introduction The renewal risk model has been playing a fundamental role in classical and modern risk theory since it was introduced by Sparre Andersen in the middle of last century as a natural generalization of the compound Poisson risk model. In the standard framework of the renewal risk model, both claim sizes X k, k ; 2; : : :, and inter-arrival times k, k ; 2; : : :, form a sequence of independent and identically distributed (i.i.d.) random variables and the two sequences are mutually independent too. Denote by X and the generic random variables Corresponding author: Qihe Tang; Tel.: ; Fax:

2 of the claim sizes and inter-arrival times, respectively. Being equipped with other modeling factors (such as initial surplus, premium incomes) and incorporated with some economic factors (such as interests, dividends, taxes, returns on investments), this model provides a good mechanism for describing non-life insurance business. It should be noted that these independence assumptions are made mainly not for practical relevance but for mathematical tractability. Among them, the one assuming complete independence between the claim size X and the inter-arrival time is especially unrealistic in almost all kinds of insurance. Think that, if the deductible retained to insureds is raised, then the inter-arrival time will increase because small claims will be ruled out, while the likelihood of a large claim will increase if X is new-worse-than-used or decrease if X is newbetter-than-used. However, it is usually challenging to get results without such independence assumptions because most of available methods will fail to work. During the last few years things have started to change. Especially in ruin theory, various non-standard extensions to the renewal risk model have been proposed to appropriately relax these independence assumptions. Initiated by Albrecher and Teugels (26), there have been a series of papers devoted to ruin-related problems of an extended renewal risk model in which (X k ; k ), k ; 2; : : :, are assumed to be i.i.d. copies of a generic pair (X; ) with dependent components X and. An advantage of this risk model is that independence between the increments of the surplus process over claim arrival times is preserved. Albrecher and Teugels (26) considered that (X; ) follows an arbitrary dependence structure, through a copula, and they derived explicit exponential estimates for nite-time and in nite-time ruin probabilities in the case of light-tailed claim sizes. Boudreault et al. (26) proposed a dependence structure that X conditional on has a density function equal to a mixture of two arbitrary density functions, and they studied the Gerber Shiu expected discounted penalty function and measured the impact of the dependence structure on the ruin probability via the comparison of Lundberg coe cients. Cossette et al. (28) assumed a generalized Farlie Gumbel Morgenstern copula to describe the dependence structure of (X; ), and they derived the Laplace transform of the Gerber Shiu function. Badescu et al. (29) assumed that (X; ) follows a bivariate phase-type distribution, and they employed the existing connection between risk processes and uid ows to the analysis of various ruin-related quantities. Very recently, Asimit and Badescu (2) introduced a general dependence structure for (X; ), via the conditional tail probability of X given, and they studied the tail behavior of discounted aggregate claims in the compound Poisson risk model in the presence of a constant force of interest and heavy-tailed claim sizes. Other non-standard extensions to the renewal risk model, not within the above-mentioned framework, can also be found in Asmussen et al. (999), Albrecher and Boxma (24, 25), and Biard et al. (28), among others. In this paper we use the same dependence structure as proposed by Asimit and Badescu (2) for the generic random pair (X; ). Precisely, we assume that the claim size X and the inter-arrival time ful ll the relation Pr (X > xj t) Pr (X > x) h(t); t ; (.) 2

3 for some measurable function h() : [; ) 7 (; ), where the symbol means that the quotient of both sides tends to as x. When t is not a possible value of, that is, Pr ( 2 ) for some open interval containing t, the conditional probability in (.) is simply understood as unconditional and, therefore, h(t). Actually, in our main results below, whenever h(t) appears it is multiplied by Pr( 2 dt). Hence, the function h(t) at such a point t can be assigned any positive value without a ecting our nal results. As discussed by Asimit and Badescu (2) (see also Section 3 below), relation (.) de nes a general dependence structure which is easily veri able for some commonly-used bivariate copulas and allows both positive and negative dependence. It also brings us great convenience in dealing with the tail behavior of the sum or product of two dependent random variables. For instance, consider the discounted value Xe r with r. If relation (.) holds uniformly for t 2 [; ), which is often the case in concrete examples, then integrating both sides with respect to Pr( 2 dt) leads to Eh(). Hence, by conditioning on we obtain Pr Xe r > x Z Pr X > xe rt h(t) Pr( 2 dt) Pr Xe r > x ; (.2) where is a random variable, independent of X, with a proper distribution given by Pr( 2 dt) h(t) Pr( 2 dt): The analysis in relation (.2) shows that the dependence structure de ned by (.) can be easily dissolved and its impact on the tail behavior of quantities under consideration can be easily captured. Consider the renewal risk model in which (X k ; k ), k ; 2; : : :, are i.i.d. copies of a generic pair (X; ) ful lling the dependence structure de ned by (.). In this paper, assuming a constant force of interest r and heavy-tailed claim sizes, we study the tail behavior of discounted aggregate claims and aim at exact asymptotic formulas. We establish local uniformity for the obtained asymptotic formulas for the subexponential case and global uniformity for the extended-regularly-varying case. More importantly, in comparison with the corresponding existing results of Tang (27) and Hao and Tang (28) in the case of independent X and, our formulas successfully capture the impact of the dependence structure of (X; ). The asymptotic behavior of the nite-time and in nite-time ruin probabilities of the renewal risk model of standard structure with a constant force of interest r > has been extensively investigated in the literature. The reader is referred to Asmussen (998), Klüppelberg and Stadtmüller (998), Kalashnikov and Konstantinides (2), Konstantinides et al. (22), Tang (25), and Wang (28). It is worthwhile noting that, if both the force of interest r > and the premium rate c are constant and the claim-size distribution is subexponential, then the tail probability of the discounted aggregate claims up to a nite or in nite time is asymptotically equivalent to the probability of ruin by the same time. This is because the amount of discounted aggregate premiums is always bounded by a nite constant cr and, thus, it does not a ect the asymptotic behavior of a subexponential tail. Due to this reason, our results in this paper can be straightforwardly translated in the form of 3

4 nite-time and in nite-time ruin probabilities. In comparison with the corresponding results of the works cited above, our formulas also explicitly show the impact of the dependence structure of (X; ) on ruin. The rest of this paper consists of four sections. Section 2 shows two main results after brie y introducing necessary preliminaries about the renewal risk model and subexponential distributions, Section 3 veri es the local and global uniformity of relation (.) through copulas, and Sections 4 and 5 prove the two theorems, respectively. 2 Main results Throughout this paper, all limit relationships hold as x unless stated otherwise. For two positive functions a() and b() satisfying l lim inf x a(x) b(x) lim sup x a(x) b(x) l 2; we write a(x) & b(x) if l, write a(x). b(x) if l 2, write a(x) b(x) if l l 2, and write a(x) b(x) if < l l 2 <. Very often we equip limit relationships with certain uniformity, which is crucial for our purpose. For instance, for two positive bivariate functions a(; ) and b(; ), we say that a(x; t). b(x; t) holds uniformly for t 2 6? if a(x; t) lim sup sup x t2 b(x; t) : Consider the renewal risk model in which (X k ; k ), k ; 2; :::, are i.i.d. copies of a generic pair (X; ) with marginal distributions F and G on [; ). To avoid triviality, both F and G are assumed not degenerate at. Denote by k P k i i, k ; 2; :::, the claim arrival times, with. Then the number of claims by time t is N t #fk ; 2; : : : : k tg; t ; which forms an ordinary renewal counting process with a nite mean function t EN t X Pr ( k t) ; t : k In this way, the amount of aggregate claims is a random sum of the form X(t) P N t k X k for t, where and throughout a summation over an empty index set produces a value. Assuming a constant force of interest r, the amount of discounted aggregate claims by time t is expressed as D r (t) e rs dx(s) X X k e r k ( k t); t ; (2.) k where the symbol E denotes the indicator function of an event E. 4

5 When studying the tail probability of D r (t) it is natural to restrict the region of the variable t to ft : < t g : With t infft : Pr ( t) > g, it is clear that [t; ] if Pr ( t) > while (t; ] if Pr ( t). For notational convenience, we write T [; T ] \ for every xed T 2. We only consider the case of heavy-tailed claim-size distributions. One of the most important classes of heavy-tailed distributions is the class S of subexponential distributions. By de nition, a distribution F on [; ) is said to be subexponential if F (x) F (x) > for all x and the relation F lim n (x) n x F (x) holds for all (or, equivalently, for some) n 2; 3; : : : ; where F n denotes the n-fold convolution of F. The class S contains a lot of important distributions such as Pareto, lognormal, and heavy-tailed Weibull distributions. See Embrechts et al. (997) for a nice review of subexponential distributions in the context of insurance and nance. A useful subclass of S is the class of distributions with extended-regularly-varying (ERV) tails, characterized by the relations F (x) > for all x and y lim inf x F (xy) F (x) lim sup x F (xy) F (x) y ; y ; (2.2) for some < <. We signify the regularity property in (2.2) as F 2 ERV( ; ), so that ERV is the union of all ERV( ; ) over the range < <. In particular, when, the class ERV( ; ) coincides with the famous class R of distributions with regularly-varying tails. Thus, if F 2 R for some < < then F (xy) lim x F (x) y ; y > : (2.3) Write R as the union of all R over the range < <. For a distribution F 2 ERV( ; ) for some < <, by Proposition of Bingham et al. (989) we know that, for every " > and b >, there is some x > such that the inequalities b y " ^ y +" F (xy) F (x) b y " _ y +" (2.4) hold whenever x > x and xy > x. In particular, when the inequalities in (2.4) reduce to the well-known Potter s bounds for the class R. Precisely, if F 2 R for some < <, then for every " > and b >, there is some x > such that the inequalities b y " ^ y +" F (xy) F (x) b y " _ y +" (2.5) 5

6 hold whenever x > x and xy > x. In addition, by Theorem.5.2 of Bingham et al. (989), the convergence in relation (2.3) is uniform for y 2 ["; ) for every xed " > ; that is to say, lim sup x y2[";) F (xy) F (x) y : (2.6) As mentioned in Section, a standing assumption on the dependence structure of (X; ) in this paper is the following: A: There is some measurable function h() : [; ) 7 (; ) such that relation (.) holds locally uniformly for t 2 (that is to say, it holds uniformly for t 2 T for every T 2 ). To achieve global uniformity of the obtained asymptotic formula, we need to strengthen Assumption A to the following: A2: There is some measurable function h() : [; ) 7 (; ) such that relation (.) holds uniformly for t 2. In addition to Assumption A or A2, we also need to assume the following: B: Either t >, or t and there is some t 2 such that inf tt h(t) >. Note that t appearing in Assumption B can be chosen to be if Pr( ) >, and in this case the restriction inf tt h(t) > is redundant since h() > by Assumption A or A2. We remark that these assumptions on the dependence structure of (X; ) are close to minimum for establishing a locally or globally uniform asymptotic formula. The rst main result of this paper is given below: Theorem 2.. Consider the discounted aggregate claims described in relation (2.) with r. If F 2 S and Assumptions A and B hold, then the relation Pr (D r (t) > x) holds locally uniformly for t 2, where F (xe rs ) d ~ s (2.7) ~ t ( + t u ) h(u)g(du): (2.8) If, in addition to the other conditions of Theorem 2., F 2 R for some < <, then applying the uniformity of relation (2.3) as explained in (2.6) to relation (2.7), we obtain that the relation Pr (D r (t) > x) F (x) e rs d ~ s (2.9) holds locally uniformly for t 2. 6

7 Consider the uniformity of relation (2.7) on T for some T 2. Under Assumption A, integrating both sides of (.) with respect to Pr( 2 dt) over the range [; T ] leads to < Eh() (T ). Similarly as in (.2), we introduce an independent random variable with a proper distribution given by Pr( 2 dt) h(t) Eh() (T ) G(dt); t 2 [; T ]: Construct a delayed renewal counting process fn t ; t g with inter-arrival times, k, k 2; 3 : : :, and a mean function t. It is easy to see that ~ t t Eh() (T ) ; t 2 T ; that is to say, ~ t is proportional to the mean function of a delayed renewal counting process whose rst inter-arrival time is a ected by the dependence structure of (X; ). Next we establish global uniformity for relation (2.7). For this purpose, we need to restrict the claim-size distribution to the class ERV. Notice that if r then D r (t) diverges to almost surely as t and, hence, it is not possible to establish the global uniformity for relation (2.7). For this reason, we assume r > in the following second main result: Theorem 2.2. Consider the discounted aggregate claims described in relation (2.) with r >. If F 2 ERV and Assumptions A2 and B hold, then relation (2.7) holds uniformly for t 2. Similarly as above, if, in addition to the other conditions of Theorem 2.2, F 2 R for some < <, then, by (2.5) and (2.6), it is easy to verify that relation (2.9) holds uniformly for t 2. In particular, taking t in relation (2.9) yields a more transparent asymptotic formula that r Eh()e Pr (D r () > x) F (x) Ee : r Moreover, under Assumption A2, which implies Eh(), ~ t is equal to the mean function of a delayed renewal counting process whose rst inter-arrival time follows Pr( 2 dt) h(t)g(dt). When X and are independent, h(t) and ~ t t for all t 2. Hence, Theorem 2. of Hao and Tang (28) corresponds to our Theorem 2. for the case of independent X and. The expression of ~ t given in (2.8) for the general case of dependent X and shows that our results successfully capture the impact of the dependence structure of (X; ) on the tail behavior of the discounted aggregate claims. Asimit and Badescu (2) studied the same problem. Our work extends theirs in the following three directions: (i) they considered the compound Poisson risk model while we consider the renewal risk model; (ii) they derived results for F 2 S when r and for F 2 R when r >, both of which are covered and uni ed by our Theorem 2.; (iii) their formulas hold for a xed time t while ours are equipped with local or global uniformity in time t, which greatly enhances the theoretical and applied interests of the results. 7

8 Restricted to the compound Poisson risk model with F 2 R for some < <, assuming the Poisson intensity > our formula (2.9) immediately gives with K r;t Pr (D r (t) > x) K r;t F (x) ; t > ; (2.) + e ru r r e rt h(u)e u du: Theorem 3.2 of Asimit and Badescu (2) also gives relation (2.) but with a coe cient X Z Z Kr;t e t n h(s i )e r P i ny j s j ds i ; n n;t where n;t f(s ; : : : ; s n ) 2 [; t] n : P n i s i tg. It is not hard to verify that the two coe cients are actually the same though they look quite di erent. Indeed, recall that, for the Poisson process fn t ; t g, the inter-arrival times, : : :, n conditional on (N t n) have a joint distribution on n;t given by i i Pr ( 2 ds ; : : : ; n 2 ds n j N t n) n t n ds ds n : Under the help of this property, we have X Kr;t (t) n Z Z e t h(s i )e r P i j s n j n t n n i n;t X Pr (N t n) E h( i )e r i N t n n X n i i Eh( i )e r i (Ntn) X Eh( i )e r i ( i t) i Thus, K r;t K r;t. h(u)e ru G(du) + X i2 h(u)e ru e u du + u u ny ds i i h(u)e r(u+v) Pr ( i 2 dv) G(du) h(u)e r(u+v) (dv) e u du : 3 Veri cation of the assumptions on dependence This section concerns veri cation of our assumptions on the dependence structure of (X; ). We carry on this discussion through copulas. The reader is referred to Joe (997) or Nelsen (26) for a comprehensive treatment of copulas. 8

9 For simplicity, assume that H, the joint distribution of (X; ), has continuous marginal distributions F and G. Then, by Sklar s theorem, there is a unique copula C(u; v) : [; ] 2 7 [; ], which is the joint distribution of the uniform variates F (X) and G(), such that H(x; t) C (F (x); G(t)) : The corresponding survival copula is de ned to be which is such that ^C(u; v) u + v + C ( u; v) ; (u; v) 2 [; ] 2 ; H(x; t) Pr (X > x; > t) ^C F (x); G(t) : Assume that the copula C(u; v) is absolutely continuous, so is the survival copula ^C(u; v). Denote by ^c(u; v) the density of the survival copula. Then the function h() de ned in (.), if it exists, is equal to h (t) lim ^C(u; u ^c +; G(t) ; t > : (3.) vg(t) In the rest of this section, the variables t and v are always connected through the identity v G(t), as indicated in (3.). In terms of the survival copula ^C(u; v), the local uniformity of relation (.), as required by Assumption A, can be restated as lim sup u+ ^C(u; uh(t) lim sup u+ v2[;] u R u ^c (s; v) ds ^c (+; v) ; 2 (; ); (3.2) and the global uniformity of relation (.), as required by Assumption A2, can be restated lim sup ^C(u; v) u+ uh(t) lim sup u ^c (s; v) ds u u+ ^c (+; v) : (3.3) v2(;] v2(;] So far, the function h() and the local/global uniformity of relation (.) have been expressed through the survival copula and its density. Recall that an Archimedean copula is of the form C(u; v) ' [ ] ('(u) + '(v)) ; (u; v) 2 [; ] 2 ; where '() : [; ] 7 [; ], called the generator of C(u; v), is a strictly decreasing and convex function with < ' () and ' (), while ' [ ] () is the pseudo-inverse of '(), equal to ' (t) when t '() and equal to otherwise. This copula has a simple structure, as its de nition shows, and it enjoys a lot of nice properties. If the generator '() is twice di erentiable, then the copula density has a transparent form; see relation (4.3.6) of Nelsen (26). In this case, recalling (3.), it follows that h(t) ^c (+; v) ' ( )' ( v) (' ( v)) 2 : 9

10 Moreover, it should not be hard to construct some general conditions on the generator '() to guarantee relations (3.2) and (3.3). Next, we reexamine the three examples given in Asimit and Badescu (2). Example 3.. The Ali-Mikhail-Haq copula is of the form Direct calculation shows C(u; v) uv ; 2 [ ; ): ( u)( ^C(u; u + u( 2v) u2 ( v 2 ) ( uv) 2 : (3.4) Then, by relation (3.), It follows uh(t) h(t) + ( 2v) : jv (2 uv) ( + ( 2v)) + v2 j ( uv) 2 jj u: ( + ( 2v)) Hence, relation (3.2) holds when 2 [ ; ) and relation (3.3) holds when 2 ( ; ). Example 3.2. The Farlie-Gumbel-Morgenstern copula is of the form C(u; v) uv + uv( u)( v); 2 [ ; ]: Going along the same lines of Example 3., we have, ^C(u; u + u ( u) ( 2v); h(t) + ( 2v) ; uh(t) j 2vj jj u: + ( 2v) Hence, relation (3.2) holds when 2 [ ; ) and relation (3.3) holds when 2 ( ; ). Example 3.3. The Frank copula is of the form C(u; v) ln + (e u ) (e v ) ; 6 : e Direct but a bit tedious calculation ^C(u; e ( e u ) e ( e ) + (e u e ) (e v e ) :

11 It follows that uh(t) h(t) e( v) e u e ( v) (e u ) + (e e u ) (e u ) (e ) : u (e ( v) (e u ) + (e e u )) Hence, relations (3.2) and (3.3) hold for all 6. 4 Proof of Theorem Lemmas It is well known that every subexponential distribution F is long tailed, denoted as F 2 L, in the sense that the relation F (x y) lim (4.) x F (x) holds for all (or, equivalently, for some) y 6 ; see Lemma 2 of Chistyakov (964) or Lemma.3.5(a) of Embrechts et al. (997). We rst show an elementary result regarding long-tailed distributions: Lemma 4.. F 2 L if and only if there is a function l() : (; ) 7 (; ) satisfying (i) l(x) < x2 for all x >, (ii) l(x), (iii) l() is slowly varying at in nity, such that, for every K >, F (x Kl(x)) F (x): Proof. The if assertion is trivial, so we only prove the only if assertion. Let F 2 L. It is easy to see that there is a positive function l () satisfying l (x), l (x) < x 2 4 for all x >, and F (x l (x)) F (x). Furthermore, by Lemma 3.2 of Tang (28), there is a slowly varying function l() : (; ) 7 (; ) satisfying l(x) and l(x) l (x) 2 for all x >. This function l() ful lls all the requirements in Lemma 4.. Lemma 4.2 below forms the main ingredient of the proof of Theorem 2.. It deals with the tail probability of the sum of n random variables equipped with a certain dependence structure. A similar problem was considered in Proposition 2. of Foss and Richards (2). However, a close look tells that their Proposition 2. and our Lemma 4.2 below are essentially di erent. Let X,..., X n be i.i.d. random variables with common distribution F 2 S. Recall Proposition 5. of Tang and Tsitsiashvili (23), which shows that, for arbitrarily xed < a b <, the relation Pr w k X k > x Pr (w k X k > x) (4.2) k k

12 holds uniformly for (w ; : : : ; w n ) 2 [a; b] n. Hence, for the case of independent X and, Lemma 4.2 below immediately follows by conditioning on ( ; : : : ; n ). However, for the general case of dependent X and, this lemma is a nontrivial consequence of Proposition 5. of Tang and Tsitsiashvili (23). Hereafter, for notational convenience, for every t 2 and n ; 2; : : :, we write t n P n i s i and n;t f(s ; : : : ; s n ) 2 [; t] n : t n tg. Lemma 4.2. Recall the renewal risk model introduced in Section 2 with r. If F 2 S and Assumption A holds, then for every n ; 2; : : :, it holds locally uniformly for t 2 that > x; N t n ( + o()) > x; N t n : (4.3) k Proof. Note that the event (N t n) in relation (4.3) could have a probability. In the proof below, we still use the notation a(x; t) b(x; t) even though the two functions a(x; t) and b(x; t) could be simultaneously equal to. For such an occasion the notation exactly means a(x; t) ( + o())b(x; t). Thus, doing so will cause no confusion. We need to prove that relation (4.3) holds uniformly for t 2 T for arbitrarily xed T 2. Let us proceed the proof by induction. Trivially, the assertion holds for n. Now we assume by induction that the assertion holds for some positive integer n m and we prove it for n m; that is to say, we aim at the relation > x; N t m > x; N t m (4.4) k with the required uniformity for t 2 T. Recall the function l() speci ed in Lemma 4.. According to the value of the sum P m k X ke r k belonging to (; l(x)], (x l(x); ), and (l(x); x l(x)], we split the probability on the left-hand side of (4.4) into three parts as > x; N t m I (x; m; t) + I 2 (x; m; t) + I 3 (x; m; t): k For I (x; m; t), it holds uniformly for t 2 T that I (x; m; t) Pr X m e r m > x l(x); N t m Z Z my Pr X m e rtm > x l(x) m s m G (t tm ) G(ds i ) i m;t Z Z my Pr X m e rtm > x h(sm )G (t t m ) G(ds i ) Z m;t Z m;t k k i Pr X m e rtm > x m s m G (t tm ) my G(ds i ) Pr X m e r m > x; N t m ; (4.5) 2 i

13 where at the third and fourth steps we used Assumption A and Lemma 4.. As it shows from the second step to the last step, the derivation of (4.5) is mainly to eliminate the slowly varying function l(x). For I 2 (x; m; t), by the induction assumption and the same idea as in deriving (4.5), we have, uniformly for t 2 T, I 2 (x; m; t) Pr k Pr k m X k m X k e r k > x k l(x); N t m X k e r k > x l(x); N t sm m G(ds m ) > x l(x); N t sm m G(ds m ) > x; N t sm m G(ds m ) X > x; N t m : (4.6) k Now we focus on I 3 (x; m; t). It holds uniformly for t 2 T that I 3 (x; m; t) + X m e rm > x; X k e r k > l(x); X m e rm > l(x); N t m k Z k k l(x) Z Pr Z l(x) Z m;t k k X k e r k > (x y) _ l(x); N t sm m Pr X m e rsm 2 dy m s m G(dsm ) > (x y) _ l(x); N t sm m Pr X m e rsm 2 dy m s m G(dsm ) Pr X k e rtk + X m e rsm > x; X k e rtk > l(x); X m e rsm > l(x) G(t t m )h(s k )h(s m ) my G(ds i ); where at the third step we used induction assumption and at the last step we applied Assumption A twice. For every k ; : : : ; m, we notice that, uniformly for (s ; : : : ; s m ) 2 m;t, Pr X k e rt k + X m e rsm > x; X k e rt k > l(x); X m e rsm > l(x) Pr X k e rt k + X m e rsm > x Pr X k e rt k > x; X m e rsm l(x) Pr X m e rsm > x; X k e rt k l(x) o() Pr X k e rt k > x + Pr X m e rsm > x ; 3 i

14 where at the last step we applied Proposition 5. of Tang and Tsitsiashvili (23) as summarized in (4.2) above. Plugging these estimates into I 3 (x; m; t), we have, uniformly for t 2 T, I 3 (x; m; t) o() k +o() Z k Z m;t Z Z m;t Pr X k e rtk > x G (t tm ) h(s k )h(s m ) my G(ds i ) i Pr X m e rsm > x G (t tm ) h(s k )h(s m ) my G(ds i ): Since the distribution F has an ultimate tail, for an arbitrary function a(x) o(), we can always nd some positive function l (x), which diverges to but slowly enough, such that a(x) o()f (l (x)) : Using this idea and Assumption A, we have, uniformly for t 2 T, Z Z my I 3 (x; m; t) o() Pr X k e rtk > x; X m > l (x) G (t tm ) h(s k )h(s m ) G(ds i ) k Z +o() k m;t Z m;t i Pr X m e rsm > x; X k > l (x) G (t tm ) h(s k )h(s m ) o() > x; X m > l (x); N t m o() k +o() Pr X m e rm > x; X k > l (x); N t m k i my G(ds i ) > x; N t m : (4.7) k A combination of (4.5), (4.6), and (4.7) gives the upper-bound version of (4.4). The corresponding lower-bound version of (4.4) can be easily established. Actually, > x; N t m Pr Pr k m X X k e r k > x [ X m e r m > x ; N t m k m X X k e r k > x; N t m + Pr X m e r m > x; N t m k m Pr X X k e r k > x; X m e r m > x; N t m k J (x; m; t) + Pr X m e r m > x; N t m J 2 (x; m; t): (4.8) 4 i

15 As in dealing with I 2 (x; m; t), by the induction assumption, we have, uniformly for t 2 T, J (x; m; t) k > x; N t m : (4.9) Furthermore, as in dealing with I 3 (x; m; t), it holds uniformly for t 2 T that J 2 (x; m; t) +X m e rm > x; X k e r k > l(x); X m e rm > l(x); N t m o() k k > x; N t m : (4.) k Plugging (4.9) and (4.) into (4.8) gives the lower-bound version of (4.4). This ends the proof of Lemma 4.2. Going along the same lines of the proof of Lemma 4.2 with some obvious modi cations, we can obtain the following result: Lemma 4.3. Under the same conditions of Lemma 4.2, for every n ; 2; : : :, it holds locally uniformly for t 2 that > x; n t ( + o()) n Pr X e r > x; n t : (4.) k For a distribution F 2 S, the well-known Kesten s inequality states that, for every " >, there is some constant K K " > such that the inequality F n (x) K( + ") n F (x) holds for all n ; 2; : : : and x. For its proof see Athreya and Ney (972) or Lemma.3.5(c) of Embrechts et al. (997). In the following lemma, we establish an inequality of Kesten s type for the probability on the left-hand side of (4.): Lemma 4.4. Recall the renewal risk model introduced in Section 2 with r. If F 2 S, t, and Assumptions A and B hold, then for every " > and T 2, there is some constant K K r;";t > such that the inequality > x; n t K( + ") n Pr X e r > x; n t k holds for all n ; 2; : : :, x, and t 2 T. Proof. For every " >, by Lemma 4.3, there is some constant x > such that, for all x > x and t 2 T, Pr X e r + X 2 e r 2 > x; X 2 e r 2 x; 2 t Pr X e r + X 2 e r 2 > x; 2 t Pr X 2 e r 2 > x; 2 t ( + ") Pr X e r > x; 2 t : (4.2) 5

16 By Assumption A, the constant x above can be chosen so large that, for all t 2 T, Pr X e r > x j t 2 F x e rt h(t): (4.3) Write Pr P n k a n sup X ke r k > x; n t : x Pr (X e r > x; n t) t2 T We start to evaluate a n+. It is clear that Xn+ > x; n+ t Pr k n+ X X k e r k > x; X n+ e r n+ x; n+ t + Pr X n+ e r n+ > x; n+ t : k Conditioning on (X n+ ; n+ ) and noticing the de nition of a n, we have Xn+ > x; X n+ e r n+ x; n+ t k a n Pr X e r + X n+ e r n+ > x; X n+ e r n+ x; n+ t : Using (4.2) we know that, for all x > x and t 2 T, Pr X e r + X n+ e r n+ > x; X n+ e r n+ x; n+ t Z Z Pr X e r + X 2 e r2 > x; X 2 e r2 x; 2 t t n n ;t Hence, ( + ") Pr X e r > x; n+ t : Pr P n+ k sup X ke r k > x; n+ t x>x Pr (X e r > x; n+ t) t2 T When x x, by inequality (4.3), it holds for all t 2 T that Pr P n+ k X ke r k > x; n+ t Pr (X e r > x; n+ t) Y G(ds i ) n i ( + ")a n + : (4.4) Pr ( n+ t) Pr (X e r > x ; n+ t) 2 F x e R t G n (t s)g(ds) rt R t G n (t s)h(s)g(ds) : By Assumption B, there is some constant t 2 such that h(s) is away from for s 2 [; t ]. We have R t R G n t^t (t s)g(ds) R t G n (t s)h(s)g(ds) G n (t s)g(ds) + R t G n (t s)g(ds) t^t (t>t ) R t^t G n (t s)h(s)g(ds) inf h(s) G(t ) + R s2[;t t ] h(s)g(ds) : 6

17 Therefore, there is some < L < such that It follows from (4.4) and (4.5) that Pr P n+ k sup X ke r k > x; n+ t xx Pr (X e r > x; n+ t) t2 T a n+ ( + ")a n + + L: L: (4.5) This recursive inequality with initial value a completes the proof of Lemma Proof of Theorem 2. We follow the proof of Theorem 2. of Hao and Tang (28) but we need to overcome some technical di culties due to the dependence structure of (X; ). Let us prove that relation (2.7) holds uniformly for t 2 T for arbitrarily xed T 2. Choose some large positive integer N and write X NX Pr (D r (t) > x) + > x; N t n I (x; t)+i 2 (x; t): (4.6) nn+ n k First, we look at I (x; t). Note that if t > then I (x; t) vanishes for some large N. Thus, we can assume t. Applying Lemma 4.4, for every " >, there is some constant K > such that, for all t 2 T, X I (x; t) > x; n t K nn+ X nn+ k ( + ") n Pr X e r > x; n t : By Assumption A, it follows that, uniformly for t 2 T, I (x; t). K K X nn+ ( + ") n F (xe rs ) h(s)g(ds) F (xe rs ) h(s ) Pr (N t s n ) G(ds ) X nn+ ( + ") n Pr (N T n ) : It is well known that the moment generating function of N T is analytic in a neighborhood of ; see, e.g. Stein (946). Thus, we may choose some " > su ciently small such that the series at the last step above converges. Therefore, for every < <, we can nd some large positive integer N such that, uniformly for t 2 T, I (x; t). F (xe rs ) d ~ s : (4.7) 7

18 Next, we turn to I 2 (x; t). By Lemma 4.2, it holds uniformly for t 2 T that I 2 (x; t) X n X > x; N t n I 2 (x; t) I 22 (x; t): (4.8) nn+ k For I 2 (x; t), by interchanging the order of the sums then conditioning on k obtain that, uniformly for t 2 T, and k, we I 2 (x; t) X > x; k t k X k u F (xe ru ) + F xe r(u+v) Pr ( k 2 dv) h(u)g(du) F (xe rs ) h(s)g(ds) + u F xe r(u+v) d v h(u)g(du) Z s F (xe rs ) d s u h(u)g(du) F (xe rs ) d ~ s ; (4.9) where at the fourth step we used integration by parts with possible jumps; see, e.g. formula (.2) of Klebaner (25). For I 22 (x; t), it holds uniformly for t 2 T that I 22 (x; t). X nn+ k X nn+ k X nn+ n > x; N t n E ( + N T ) (NT N) F (xe rs k ) Pr (N t sk n ) h (s k ) G(ds k ) F (xe rs ) Pr (N t s n ) h (s) G(ds) F (xe rs ) h (s) G(ds): Hence, we can nd some large positive integer N such that, uniformly for t 2 T, I 22 (x; t). F (xe rs ) d ~ s : (4.2) Plugging (4.9) and (4.2) into (4.8) yields that, uniformly for t 2 T, ( ) F (xe rs ) d ~ s. I 2 (x; t). F (xe rs ) d ~ s : (4.2) Plugging (4.7) and (4.2) into (4.6) and noticing the arbitrariness of, we complete the proof of Theorem 2.. 8

19 5 Proof of Theorem Lemmas The following lemma describes closure of the class ERV under the product of two dependent random variables: Lemma 5.. Recall the renewal risk model introduced in Section 2 with r. If F 2 ERV( ; ) for some < < and Assumption A2 holds, then for every k ; 2; : : :, the distribution of X k e r k still belongs to ERV( ; ) and Proof. For every k ; 2; : : :, conditioning on k, k have > x > x F (x): (5.) Z Z It follows from (5.2) and (2.2) that, for every y, and lim sup x lim inf x Pr (X k e r k > xy) lim sup Pr (X k e r k > x) x and recalling Assumption A2, we F xe r(u+v) h(u)g(du) Pr ( k 2 dv) : (5.2) sup s F (xye rs ) F (xe rs ) y Pr (X k e r k > xy) lim inf Pr (X k e inf F (xye rs ) r k > x) x s F (xe rs ) y : Therefore, the distribution of X k e r k belongs to ERV( ; ). Moreover, applying (2.4) to (5.2), we obtain relation (5.). The following lemma is interesting in its own right and it will be the main ingredient of the proof of Theorem 2.2: Lemma 5.2. Recall the renewal risk model introduced in Section 2 with r >. If F 2 ERV( ; ) for some < < and Assumption A2 holds, then it holds uniformly for n ; 2; : : : that > x > x : (5.3) k Proof. By Lemma 5., the distributions of X k e r k, k ; 2; : : :, belong to ERV( ; ) and, for i 6 j n, Pr X i e r i > x; X j e r j > x F (x) 2 o Pr Xi e r i > x + Pr X j e r j > x : Hence, it follows from Theorem 3. of Chen and Yuen (29) that relation (5.3) holds for every xed n ; 2; : : :. Now, we turn to the required uniformity of relation (5.3). Trivially, it holds for every k n + ; n + 2; : : : that X k e r k Xk e r k, where Xk and e r k on the right-hand side 9 k

20 are independent. Hence, going along the same lines of the proof of Theorem 3. of Tang and Tsitsiashvili (24) with some obvious modi cations (see also Section 4 of Chen and Ng 27), we obtain lim lim sup n x F (x) Pr X kn+ X k e r k > x lim n lim sup x X kn+ Pr (X k e r k > x) F (x) This means that, for every >, there is some large positive integer n such that X X > x + > x. F (x): (5.4) kn + kn + By relation (5.3) with n n, the rst assertion of Lemma 5., and relation (5.4), it holds for arbitrarily xed < " < and uniformly for n > n that Xn X > x > ( ") x + > "x k k kn + Xn. ( ") > x + " F (x) k : ( ") > x + " F (x): (5.5) Symmetrically, it holds uniformly for n > n that Xn > x > x k k & k k kn + > x > x F (x): (5.6) k By relation (5.) and the arbitrariness of " and, we conclude from (5.5) and (5.6) that relation (5.3) holds uniformly for n > n. The uniformity of relation (5.3) for n n is obvious since it holds for every xed n ; 2; : : :. This completes the proof of Lemma Proof of Theorem 2.2 Following the proof of Lemma 4.2 of Hao and Tang (28), we have lim lim sup t x R t R t F (xe rs )d ~ s F (xe rs )d ~ : s 2

21 Thus, for every >, there is some T 2 such that Z T F (xe rs ) d ~ s. Z T F (xe rs ) d ~ s : (5.7) On the one hand, by Theorem 2. and relation (5.7), it holds uniformly for t 2 (T ; ] that Pr (D r (t) > x) Pr (D r (T ) > x) F (xe rs ) d ~ s & ( ) T Z t On the other hand, by Lemma 5.2 and Assumption A2, Pr (D r () > x) X > x k Pr X e r > x + Z Z Z F (xe rs ) h (s) G(ds) + F (xe rs ) d ~ s : (5.8) > xe rv d v Z Z F xe r(u+v) h(u)g(du)d v F (xe rs ) d ~ s : (5.9) Hence, by (5.9) and (5.7), it holds uniformly for t 2 (T ; ] that Pr (D r (t) > x) Pr (D r () > x) Z + F (xe rs ) d ~ s. ( + ) t F (xe rs ) d ~ s : (5.) By the arbitrariness of in (5.8) and (5.), we prove the uniformity of (2.7) for t 2 (T ; ]. Recall that Theorem 2. already shows the local uniformity of (2.7). This ends the proof of Theorem 2.2. Acknowledgment. The authors are most grateful to the two referees for their very thorough reading of the paper and valuable suggestions. The research of Jinzhu Li and Rong Wu was supported by the National Basic Research Program of China (973 Program, Project No: 27CB8495). References [] Albrecher, H.; Boxma, O. J. A ruin model with dependence between claim sizes and claim intervals. Insurance Math. Econom. 35 (24), no. 2,

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