Scandinavian Actuarial Journal. Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks

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1 Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks For eer Review Only Journal: Manuscript ID: SACT-- Manuscript Type: Original Article Date Submitted by the Author: -Sep- Complete List of Authors: Yang, Yang; Nanjing Audit University, School of Mathematics and Statistics Konstantinides, Dimitrios; University of the Aegean, Statistics Keywords: Asymptotics, consistent variation, financial and insurance risks, finite and infinite time ruin probabilities, dependence

2 age of Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks For eer Review Only Yang Yang, and Dimitrios G. Konstantinides. School of Mathematics and Statistics, Nanjing Audit University, Nanjing, China. School of Economics and Management, Southeast University, Nanjing, China. Department of Mathematics, University of the Aegean, Karlovassi, Greece Abstract Let us consider a discrete-time insurance risk model with insurance and financial risks, where the net insurance loss within period i and the stochastic discount factor from time i to time i follow some certain dependence structure for each fixed i. Under the assumption that the distribution of net insurance loss within one time period is consistently-varying-tailed, precise estimates for finite and infinite time ruin probabilities are derived. Furthermore, these estimates are uniform with respect to the time horizon. Keywords: Asymptotics; consistent variation; financial and insurance risks; finite and infinite time ruin probabilities; dependence Mathematics Subject Classification: ; E; B Introduction In this paper, we consider a discrete-time insurance risk model with financial and insurance risks in a stochastic economic environment, which was proposed by Nyrhinen,. In this environment, within period i, i, the insurer s net insurance loss i.e. the total claim amount minus the total premium income is denoted by a real-valued random variable r.v. X i. Suppose that the insurer invests his/her surplus into both risk-free and risky assets, which lead to an overall positive stochastic discount factor Y i from time i to time i. In the terminology of Norberg, r.v.s {X i, i } and {Y i, i } are called insurance risks and financial risks, respectively. For each positive integer n, the sum S n = X i Y j,. represents the stochastic discounted value of aggregate net losses up to time n. In this paper we are interested in the finite time ruin probability by time n and the infinite time ruin probability, Research supported by National Natural Science Foundation of China No.,, China ostdoctoral Science Foundation No. M, Natural Science Foundation of Jiangsu rovince of China No. BK, Qing Lan roject. Corresponding author. konstant@aegean.gr

3 age of which are defined, respectively, by ψx, n = max S k > x,. k n ψx = lim ψx, n = sup S n > x,. n n For eer Review Only where x is interpreted as the initial wealth of the insurer. Many works have been devoted to the investigation of the asymptotic behavior of ruin probabilities in the above discrete-time insurance risk model, assuming that both {X i, i } and {Y i, i } are sequences of independent and identically distributed i.i.d. r.v.s and they are independent of each other as well. One can be referred to Tang and Tsitsiashvili a,, Hashorva et al. among others. Clearly, if we denote the product i Y j in. by a weight r.v. θ i, then the investigation on ruin probabilities ψx, n and ψx boils down to the study of the asymptotics for the tail probabilities of the maximum of randomly weighted sums. Following the work of Tang and Tsitsiashvili b, there has been a vast amount of literature in this aspect, such as Goovaerts et al., Wang et al., Wang and Tang. We remark that the assumption of complete independence is for the mathematical convenience, but appears unrealistic in most practical situations. A recent new trend of the study is to introduce various dependence structures to the risk model. One direction is to allow some dependent {X i, i } and arbitrarily dependent {θ i, i }, but keep the independence between {X i, i } and {θ i, i }, see Zhang et al., Chen and Yuen, Shen et al., Weng et al., Gao and Wang, Yi et al. among others. In the presence of heavy-tailed insurance risks, they established the asymptotic formula ψx, n holds for each fixed n, or X i θ i > x = ψx X i Y j > x, x,. X i θ i > x, x.. In such a type of dependent risk model, Shen et al. obtained the asymptotics for the finite time ruin probability is uniform with respect to the time horizon i.e. relation. holds uniformly for all n, hence apply for the case of infinite time ruin i.e. relation. holds. Recently, Chen considered another type of the discrete-time risk model by allowing the dependence between insurance and financial risks. recisely speaking, she assumed that {X i, Y i, i } form a sequence of i.i.d. random vectors with generic random vector X, Y following a bivariate Farlie-Gumbel-Morgenstern distribution. Under the condition that the net loss insurance risk distribution is subexponential, she derived the asymptotic formula. for the finite time ruin probability. Some related results can be found in Yang et al. a and Yang and Hashorva. Motivated by Chen, in the present paper we aim to allow the generic random vector X, Y following a more general dependence structure. We firstly derive two general asymptotic formulas for the ruin probabilities in. and. under the condition that the loss distribution is consistently-varying-tailed see the definition in Section. In doing so, each pair of the net loss insurance risk and stochastic discount factor financial risk follows Assumptions A A in Section. Then we investigate the uniformly asymptotic behavior of the ruin probabilities with respect to n. Finally, we pursue some refinements of the general formulas by restricting the loss distribution to be regularly-varying-tailed. The obtained formulas are further refined

4 age of to be completely transparent. All estimates obtained for ruin probabilities successfully capture the impact of the underlying dependence structure of X, Y. The rest of this paper is organized as follows. Section prepares some preliminaries of heavy-tailed distributions and dependence structures, Section presents the main results of the present paper and Section gives their proofs. reliminaries For eer Review Only Throughout the paper, all limit relationships hold for x tending to unless stated otherwise. For two positive functions ax and bx, we write ax bx if lim ax/bx = ; write ax = O[bx] if lim sup ax/bx < ; write ax bx if ax = O[bx] and bx = O[ax]; and ax = o[bx] if lim ax/bx =. Furthermore, for two positive bivariate functions ax, t and bx, t, we write ax, t bx, t uniformly for all t in a nonempty set A, if lim sup ax, t x bx, t =. t A The indicator function of an event A is denoted by I A. For real x and y, denote by x y = max{x, y}, x y = min{x, y} and x = x. C represents a positive constant, whose value may vary from place to place. We shall restrict the net loss insurance risk distribution to some classes of heavy-tailed distributions. An important class of heavy-tailed distributions is D, which consists of all distributions with dominated variation. A distribution F = F on R belongs to the class D, if lim sup F x y/f x < for any < y <. A slightly smaller class is C of consistently-varyingtailed distributions. A distribution F on R belongs to the class C, if lim y lim inf F x y/f x =. Closely related distribution class is the class L of long-tailed distributions. A distribution F on R belongs to the class L, if F x y F x for any y >. A useful subclass of the class C is the class of regularly varying tailed distributions, denoted by R. A distribution F on R belongs to the class R α, if lim F x y/f x = y α for some α and all y >. It is well known that the following inclusion relationships hold: R α C L D L see, e.g., Cline and Samorodnitsky. Furthermore, for a distribution F on R, denote its upper and lower Matuszewska indices, respectively, by J F = lim y log F y log y J F = lim log F y y log y F x y with F y := lim inf F x with F F x y y := lim sup F x for y >, for y >. Clearly, F D is equivalent to J F <. For more details, see Bingham et al., Chapter.. The following property of the class D is due to Lemma. of Tang and Tsitsiashvili a. Lemma.. For a distribution F D on R, x p = o[f x] holds for any p > J F. We next introduce the dependence structure between the net loss and the stochastic discount factor via some restriction on their copula function. Let X, Y be a random vector with continuous marginal distributions F x and Gy, then the dependence structure of X and Y is characterized in terms ofurl: a bivariate copula function Cu, v by the Sklar representation

5 age of see Nelsen or Joe, and the joint distribution of X and Y can be given by C[F x, Gy]. Define the corresponding survival copula function by Ĉu, v = u v Cu, v, u, v [, ]. Assume that the copula function Cu, v is absolutely continuous. Denote by C u, v = u Cu, v, C u, v = v Cu, v, C u, v = u v Cu, v, Ĉu, v = v Ĉu, v = C u, v and Ĉu, v = u v Ĉu, v = C u, v. Recently, in the study of the closure and tail behavior of the product of two dependent r.v.s, Yang and Sun utilized the following two assumptions on the absolutely continuous copula function Cu, v. We remark that Assumption A can be attributed to Albrecher et al. and A to Asimit and Badescu. For eer Review Only Assumption A. There exists a positive constant M such that Assumption A. The relation holds uniformly for all v, ]. lim sup lim sup C u, v < M. v u Ĉ u, v u Ĉ, v, u, Remark.. Clearly, Assumption A is equivalent to C u, v u C u, v, u, holds uniformly for all v [,. Thus, if the copula Cu, v of random vector X, Y satisfies Assumptions A and A, then the copula of X, Y, denoted by C u, v, satisfies these two assumptions as well. Similar assumptions related to Assumption A can be found in Ko and Tang. ointed out by Yang and Sun, some commonly used copula functions satisfy Assumptions A and A such as the Ali-Mikhail-Haq copula of the form u v Cu, v =, r,, r u v the Farlie-Gumbel-Morgenstern copula Cu, v = u v r u v u v, r,, and the Frank copula Cu, v = r log eru e rv e r, r. In order to derive our main results, we further need the third assumption below. Assumption A. The relation holds uniformly for all v [, ]. C u, v, u, It is easy to verify that Assumption A is satisfied by the above three copula functions. We remark that such an assumption is equivalent to the fact that X x Y = y = C [F x, Gy], x,. holds uniformly for all y R.

6 age of Main results Recall the discrete-time insurance risk model described in Section. In the sequel, denote by F on R, G on R and H on R the distributions of the net loss X, the stochastic discount factor Y and their product X Y, respectively. For each i, denote by H i the distribution of X i i Y j, j. Clearly, H = H. Now we state the main results of the paper. The first two theorems derive some asymptotics for the finite and infinite time ruin probabilities, respectively. For eer Review Only Theorem.. In the discrete-time risk model described in Section, assume that {X i, Y i, i } are i.i.d. random vectors with generic random vector X, Y satisfying Assumptions A A. If F C and EY p < for some p > J F, then, for each fixed n, it holds that ψx, n H i x.. Theorem.. Under the conditions of Theorem., if J F > and EY p < for some p > J F, then it holds that ψx H i x.. The third result below shows that the asymptotic relation. for the finite time probability is uniform with respect to n. Theorem.. Under the conditions of Theorem.,. holds uniformly for all n. We now return to an important special case of the above three theorems in which we assume that F R α for some α. By using the well-known Breiman theorem see Breiman and Cline and Samorodnitsky, we focus on improving the asymptotic formulas given in. and. to be completely transparent. Define a nonnegative r.v. Y c, independent of {X, X i, i } and {Y, Y i, i }, with the distribution G c, given by G c dy = C [, Gdy] = C [, Gy]Gdy.. Since Y is a positive r.v., by Assumption A we know that Y c is not degenerate at zero. Corollary.. Under the conditions of Theorem., if F R α for some α, then, for each fixed n, it holds that ψx, n EY α c EY α n F x,. EY α by convention, EY α n / EY α = n if EY α =. Under the conditions of Theorem., if F R α for some α >, then ψx Moreover,. holds uniformly for all n. EY c α F x.. EY α

7 age of roofsfor eer We start with some lemmas. The first one presents a property of the distribution with dominated variation, which is due to Zhou et al. see rop... Lemma.. A distribution F D if and only if for any distribution G on R satisfying Gx = o[f x], there exists a positive function g such that gx, x gx and G[x gx] = o[f x]. The next two lemmas are related to the product of two r.v.s. Lemma.. Let X be a real-valued r.v. with distribution F, and Y be a nonnegative and nondegenerate at zero r.v., independent of X, with distribution G. If F C and Gx = o[f x], then the distribution H of the product X Y belongs to the class C as well. Review Only roof. By Lemma. iii of Yang et al. b, we have that for any fixed y, ] which, combined with F C, implies H C. Hx y F x y lim sup lim sup Hx F x, Lemma.. Let X be a real-valued r.v. with distribution F, and Y be a positive r.v. with distribution G. Assume that X and Y are dependent according to the copula function Cu, v satisfying Assumptions A and A. If there exists a positive function g such that then gx, x gx and G[x gx] = o[hx],. Hx X Y c > x,. where Y c is a nonnegative and non-degenerate at zero r.v., independent of X, with distribution defined in.. If F D and Gx = o[f x], then. holds. roof. The proof of this part of the lemma can be found in Yang and Sun, see Lemma. of that paper. Note that Y c is a nonnegative and non-degenerate at zero r.v., then there exists a positive constant a such that Y c > a >. By Assumption A we have that Hx X Y > x, Y > a = a x ] Ĉ [F, Gu Gdu a x F a a a X > x a Y = u Gdu x C [, Gu] Gdu = Y c > a F, a which, combined with F D, implies F x = O[Hx]. According to Lemma., there exists a positive function g such that. is satisfied. Therefore, the desired. follows from statement. In the following lemma from akes, we find a sufficient condition on the equivalence of two differences.

8 age of Lemma.. Let f i, i =,,,, be four positive functions. If f x = O[f x], f x = O[f x], f x = O[f x] and f x f x f x f x, then it holds that f x f x. roof of Theorem.. Clearly, for each n X i Y j > x ψx, n For eer Review Only which indicates that the desired. holds if we can establish and X i X i Y j > x Y j > x X i Y j > x, H i x,. H i x.. We deal with relation.. Since {X i, Y i, i } are i.i.d. random vectors, it holds that X i d Y j = n X i j=i Y j =: T n, n, where d = stands for equality in distribution. Thus, for., it is sufficient to prove T n > x H i x.. We will prove F Tn C and. by induction on n. Since EY p < for some p > J F, then by Markov inequality and Lemma. we know Gx x p EY p = o[f x],. which, according to Lemma., implies that. holds. Recall that Y c is a nonnegative and non-degenerate at zero r.v., independent of {X, X i, i } and {Y, Y i, i }, with distribution defined in.. As done in the proof of Theorem. of Yang and Sun, by Assumption A, lim sup G cx Gx = lim sup C [, Gx] Gx = lim sup u lim sup C u, v < M,. v which, combined with., implies G c x = o[f x]. Thus, by., F C and Lemma., T = X Y has the distribution H C and. trivially holds for n =. We aim to prove F Tn C and. holds for n, by assuming that F Tn C and. holds for n. For each i, since X i Y i is independent of Y,..., Y i, by E i Y j p = EY p i < for some p > J F and H C, as done in., we can get that H i C. In addition, by., F C D and EY p < hence, EYc p < by. for some p > J F, according to Theorem. iv of Cline and Samorodnitsky, we have Hx F x. This, together with. and Lemma., yields that there exists a positive function g such that. holds. By H i C L, i =,..., n, there exists a positive function hx such that g x URL: hx and H i [g x u] H i [g x].

9 age of holds uniformly for all u hx. For each n, since X n, Y n is independent of T n, we divide the tail probability T n > x into four parts: T n > x = T n X n Y n > x hx xgx = Clearly, by., hx hx xgx For eer Review Only xgx hx xgx T n > x v u X n du, Y n dv =: I I I I.. By the inductive assumption.,. and., we have I = = hx xgx hx hx xgx hx I G[x gx] = o[hx].. H i x v u X n du, Y n dv x H i X n du, Y n dv v X i Y j Y i > x, hx < X i hx, Y i x gx n H i x i= X i Y j Y i > x, X i > hx, Y i xgx X i Y j Y i > x, X i hx, Y i x gx o[hx] =: I I I o[hx].. By Assumption A i.e.., we have I = = o xgx X i Y j > x X i hx Y i = u Gdu u [ X i Y j > x n ] Gdu = o H i x.. u x gx i=

10 age of As for I, Since hx, by Assumptions A, A and., we obtain I = T n X n Y n > x, X n > hx, Y n xgx xgx = X n > hx x v u Yn = v Gdv T n du xgx = Ĉ [F hx x ] v u, Gv Gdv T n du xgx F hx x v u C [, Gv] Gdv T n du = F hx x v u C [, Gv] Gdv T n du o[hx] For eer Review Only = X n T n Y c > x, X n > hx o[hx] = X n T n Y c > x X n T n Y c > x, hx < X n hx. X n T n Y c > x, X n hx o[hx] =: I I I o[hx]. For each i, since X i Y i is independent of Y,..., Y i, by Lemma. iii of Yang et al. b, it holds that for any y >, [H i ] y = lim inf H ix y H i x Hx y lim inf Hx = lim inf X Y c > x y X Y c > x lim inf F x y F x = F y, which implies that for each i, J H i J F. For each i, this, together with H i C and < by EY p < and. for some p > J F J H i, yields that EY p c X i Y j Y c > x H i x,. according to Theorem. iv of Cline and Samorodnitsky. By Assumption A and. we have G c x gx = xgx C [, Gu] Gdu = o[hx].. Since X n, T n and Y c are mutually independent, then, by the inductive assumption.,.,. and., we can get I = hx xgx hx xgx xgx xgx = o T n > x v u G c dv F du O[G c x gx] H i x v u G c dv F du o[hx] H i x v hx G c dv F hx o[hx] H i x v X i G c dv F hx o[hx] [ n Y j Y c > x o[hx] = o H i x ]..

11 age of We next deal with I. Similarly to., by Assumption A, the inductive assumption. and., we have I = T n X n Y n > x, X n hx, Y n xgx T n hx Y n > x, X n hx, Y n x gx xgx = T n > x u hx X n hx Y n = u Gdu [ xgx x n ] = o H i Gdu = o H i x.. u For eer Review Only lugging..,. and. into., we obtain T n > x I I I I o [ n i= ] H i x =: J J o [ n ] H i x.. Since X n and T n are independent, by using Theorem. of Cai and Tang, together with F C, the inductive assumptions F Tn C and.,. and., we obtain J = n xgx H i x i= n H i x i= n H i x xgx xgx X n T n > x G c du u X n > x u x H i u G c du o[hx] X i Y j Y c > x.. To estimate J, as done in., by., Assumption A, the inductive assumption.,. and., we derive J = X i Y j > x X i > hx Y i = u Gdu u hx xgx hx = F [ hx] hx xgx T n > x v u G c dv F du o[hx] X i Y j > x u hx H i x v X i F [hx] C [, Gu] Gdu G c dv F du o[hx]. Y j Y c > x o[hx] X i Y j Y c > x. In Lemma., let consider f x = J J, f x = J, f x = n H ix and f x = n X i i Y j Y c > x. Then, by. and., clearly, f x f x = J f x

12 age of f x and f x f x; and by., lim sup f n x lim sup X i i Y j Y c > x f x n H ix n X i i lim sup Y j Y c > x <,. H i x For eer Review Only which implies f x = Of x. Next, we find that f x = Of x. Indeed, by. and., we have lim sup f x f x lim sup J f x = lim sup f x f x <. Thus, all conditions in Lemma. are satisfied, hence, combined with., it holds that [ n ] n T n > x J J o H i x H i x. This shows that the desired. holds for n. And by H i C, i, it is easy to see F Tn C. It ends the proof of.. Finally, we turn to relation.. Denote by T n = n n X i j=i Y j. The proof is same to that of. by noting Remark., and replacing.,.,., respectively, by hx xgx xgx T n > x = where I n H i x i= hx xgx T n > x v u X n du, Y n dv =: I I I, =: I I o[hx], X i Y j Y i > x, X i > hx, Y i x gx o[hx] I X n T n Y c > x X n T n Y c > x, X n hx o[hx] =: I I o[hx], dropping the redundant.,.,.; and substituting X i, T i with X i, T i, respectively, i, in all relations in the above proof. The next two lemmas will be needed in the proof of Theorem., and they are found in Yi et al.. Lemma.. Let X be a real-valued r.v. with distribution F D and J F >, and Y be a nonnegative and non-degenerate at zero r.v., independent of X. Then, for any fixed constants < p < J F J F < p <, there exist two positive constants C and x, which are irrespective of Y, such that for all x x, X Y > x C F x EY p EY p. Lemma.. Let X and Y be two independent nonnegative r.v.s. If F D, the distribution of X, and Y is not degenerate at zero, then, for any fixed p > J F, there exists a positive constant C, which is irrespective of Y, such that for all x, URL: EX Y p I {X Y x} C x p X Y > x.

13 age of roof of Theorem.. For any < p < J F J F < p <, denote by µ p = EY p EY p. Then, by Jensen inequality, EY p < implies that µ p <. We will prove relation. by using the method of the tail asymptotics for randomly weighted sums. Let Z i = X i Y i, i, Θ = and Θ i = i Y j, i. Clearly, for each i, Z i is independent of Θ i. By Lemma., there exist two positive constants C and x, irrespective of Θ i, i, such that for each i and all x x, For eer Review Only H i x = Z i Θ i > x C Hx EΘ p i EΘ p i = C µ i p Hx.. For the lower bound of., by Theorem. and., we have that for any fixed n, ψx lim inf H ix ψx, n H i x lim inf n H lim sup ix Hx i=n C µ i p as n.. i=n Now we deal with the upper bound of.. For any < δ < and any n, we have ψx X i X i Y j > x Y j > δ x i=n X i Y j > δ x =: K x K δ x.. From. and H i C, i, we obtain that for any n K x n lim sup H lim sup H i δ x n ix H ix n H i δ x lim sup as δ.. H i x For each i, define two independent r.v.s Z i = X i Y i and Θ i = i Y j. Then, by Markov inequality we have K x i=n i=n Z i Θ i > x H i x x p E i=n i=n Z i Θ i I {Z i Θ i x} > x Z i Θ i I {Z i Θ i x} p =: K x K x.. If p, then, by using the inequality x j xq j /q with x j, j and < q, and according to Lemma., there exists a positive constant C, irrespective of Θ i, i, such that for all x > K x x p i=n EZ i Θ i p I {Z i Θ i x} C i=n Z i Θ i > x = C URL: K x

14 age of If p >, then by Minkowski inequality and Lemma., we have p K x x p EZ i Θ i p p I {Z i Θ i x} i=n p C H i x p.. For eer Review Only i=n i=n Combining.. and by., µ p <, we obtain lim sup K x Hx C p µ i i p p µ p as n, which, together with H C D, implies i=n K δ x lim sup H ix lim sup K δ x Hδ x lugging. and. into., we can derive lim sup Therefore, the desired. follows from. and.. lim sup Hδ x Hx =.. ψx H ix.. roof of Theorem.. We firstly estimate the upper bound. For any positive x and any N { N ψx, n sup n H ix max ψx, n } n H ix, ψx n H.. ix n By Theorem. we have lim n= N n= By Theorem.,. and µ p < we have lim sup n=n ψx n H ix n=n ψx, n n H ix =.. ψx lim H lim sup ix i=n C i=n µi p C i=n µi p Then, the upper bound follows from... For the lower bound, similarly, for any positive x and any N, inf n By Theorem. we have { N ψx, n n H ix min n= ψx, n n H ix, n=n H i x i=n Hx H i x Hx as N.. } ψx, N n H ix.. N ψx, n lim n H ix =.. n=

15 age of By Theorem.,. and µ p < we have lim inf n=n ψx, N n H ix lim C ψx, N N H ix lim inf For eer Review Only i=n Therefore, the lower bound is obtained from... i=n H i x Hx µ i p as N.. roof of Corollary.. We firstly prove part. If F R α and EY p < hence, by. holds EYc p < for some p > J F = α, then, by. in Lemma. and Breiman theorem, we have Hx X Y c > x EYc α F x, which implies H R α. Again by Breiman theorem, we obtain that for each i H i x EY α c EY α i F x.. Therefore, the desired. follows from. and.. The proof of part is similar to that of part. The claims follow from. and by using Theorems. and., respectively. References [] Albrecher, H., Asmussen, S. and Kortschak, D.,. Tail asymptotics for the sum of two heavy-tailed dependent risks. Extremes,. [] Asimit, A. V. and Badescu, A. L.,. Extremes on the discounted aggregate claims in a time dependent risk model. Scand. Actuar. J.,. [] Bingham, N. H., Goldie, C. M. and Teugels, J. L.,. Regular Variation. Cambridge University ress, Cambridge. [] Breiman, L.,. On some limit theorems similar to the arc-sin law. Theory robab. Appl.,. [] Cai, J. and Tang, Q.,. On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. robab.,. [] Chen, Y.,. The finite-time ruin probability with dependent insurance and financial risks. J. Appl. robab.,. [] Chen, Y. and Yuen, K. C.,. Sums of pairwise quasi-asymptotically independent random variables with consistent variation. Stochastic Models,. [] Cline, D. B. H. and Samorodnitsky, G.,. Subexponentiality of the product of independent random variables. Stochastic rocess. Appl.,. [] Gao, Q. and Wang, Y.,. Randomly weighted sums with dominated varying-tailed increments and application to risk theory. J. Korean Stat. Society,.

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