Advances in Decision Sciences

Volume 2012 (2012), Article ID 936525, 17 pages

doi:10.1155/2012/936525

Research Article

Uniform Estimate of the Finite-Time Ruin Probability for All Times in a Generalized Compound Renewal Risk Model

Qingwu Gao,¹ Na Jin,² and Juan Zheng³

¹School of Mathematics and Statistics, Nanjing Audit University, Nanjing 211815, China

²Investment Department, Nanjing Times Media Co., Ltd., Nanjing 210039, China

³School of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, China

Received 4 April 2012; Accepted 23 September 2012

Academic Editor: Raymond Chan

Copyright © 2012 Qingwu Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We discuss the uniformly asymptotic estimate of the finite-time ruin probability for all times in a generalized compound renewal risk model, where the interarrival times of successive accidents and all the claim sizes caused by an accident are two sequences of random variables following a wide dependence structure. This wide dependence structure allows random variables to be either negatively dependent or positively dependent.

1. Introduction

In this section, we will introduce a generalized compound renewal risk model, some common classes of heavy-tailed distributions, and some dependence structures of random variables (r.v.s), respectively.

1.1. Risk Model

It is well known that the compound renewal risk model was first introduced by Tang et al. [1], and since then it has been extensively investigated by many researchers, for example, Aleškevičienė et al. [2], Zhang et al. [3], Lin and Shen [4], Yang et al. [5], and the references therein. In the paper, we consider a generalized compound renewal risk model which satisfies the following assumptions.

Assumption

The interarrival times of successive accidents are nonnegative, identically distributed, but not necessarily independent r.v.s with finite mean .

Assumption

The claim sizes and their number caused by th accident are and , , respectively, where are nonnegative and identically distributed r.v.s with common distribution and finite mean , and are not necessarily independent r.v.s, but and are mutually independent for all , , while are independent, identically distributed (i.i.d.), and positive integer-valued r.v.s with common distribution and finite mean .

Assumption

The sequences , , and are mutually independent.

Denote the arrival times of the th accident by , , which can form a nonstandard renewal counting process

with mean function . Hence the total claim amount at time and the total claim amount up to time are, respectively,

and then the insurer’s surplus process is given by

where is the initial surplus and is the constant premium rate. The finite-time ruin probability within time is defined as

Clearly, the ruin can only arise at the times , , then

Let be a nonnegative r.v., the random time ruin probability is

In order for the ultimate ruin not to be certain, we assume the safety loading condition holds, namely,

In the generalized compound renewal risk model above, if all the sequences , , and are i.i.d. r.v.s, then the model is reduced to the standard compound renewal risk model introduced by Tang et al. [1], if , then the model is the renewal risk model, see Tang [6], Leipus and Šiaulys [7], Yang et al. [8], and Wang et al. [9], among others.

1.2. Heavy-Tailed Distribution Classes

We now present some common classes of heavy-tailed distributions. Firstly, we introduce some notions and notation. All limit relationships in the paper are for unless mentioned otherwise. For two positive functions and , we write if , write if , write if both, write if . For two positive bivariate functions and , we say that relation holds uniformly for all if

For a distribution on , denote its tail by , and its upper and lower Matuszewska indices by, respectively, for ,

where and .

Chistyakov [10] introduced an important class of heavy-tailed distributions, the subexponential class. By definition, a distribution on belongs to the subexponential class, denoted by , if

where denotes the 2-fold convolution of . Clearly, if then is long tailed, denoted by and characterized by

One can easily see that a distribution if and only if there exists a function such that

Korshunov [11] introduced a subclass of the class , the strongly subexponential class, denoted by . Say that a distribution , if and

holds uniformly for , where with an indicator function of set . Feller [12] introduced another important class of heavy-tailed distributions, the dominant variation class, which is not mutually inclusive with the class . Say that a distribution on belongs to the dominant variation class, denoted by , if

Cline [13] introduced a slightly smaller class of , the consistent variation class, denoted by . Say that a distribution if

Specially, the class covers a famous class , called the regular variation class. By definition, a distribution , if there exists some such that

It is well known that for the distributions with finite mean, the following inclusion relationships hold properly, namely,

see, for example, Cline and Samorodnitsky [14], Klüppelberg [15], Embrechts et al. [16], and Denisov et al. [17]. For more details of heavy-tailed distributions and their applications to finance and insurance, the readers are referred to Bingham et al. [18] and Embrechts et al. [16].

1.3. Wide Dependence Structure

In this section we will introduce some concepts and properties of a wide dependence structures of r.v.s, which was first introduced by Wang et al. [19] as follows.

Definition 1.1. Say that r.v.s are widely upper orthant dependent (WUOD), if for each , there exists some finite positive number such that, for all , ,

Say that r.v.s are widely lower orthant dependent (WLOD), if for each , there exists some finite positive number such that, for all , ,

Furthermore, are said to be widely orthant dependent (WOD) if they are both WUOD and WLOD.

The WUOD, WLOD, and WOD r.v.s are collectively called as widely dependent r.v.s. Recall that if or for each in Definition 1.1, then are negatively upper orthant dependent or negatively lower orthant dependent (NLOD), see Ebrahimi and Ghosh [20] or Block et al. [21]; if for some constant and each such that the two inequalities in Definition 1.1 both hold, then are extended negatively dependent, see Liu [22] and Chen et al. [23]. Obviously, the WUOD and WLOD structures allow a wide range of negative dependence structures among r.v.s, such as extended negative dependence, negatively upper orthant dependence/negatively lower orthant dependence, negative association (see Joag-Dev and Proschan [24]), and even some positive dependence. For some examples to illustrate that the WUOD and WLOD structures allow some negatively and positively dependent r.v.s, we refer the readers to Wang et al. [19].

The following properties for widely dependent r.v.s can be obtained immediately below.

Proposition 1.2. Let be WLOD (or WUOD). If are nondecreasing, then are still WLOD (or WUOD); if are nonincreasing, then are WUOD (or WLOD).

If are nonnegative and WUOD, then for each ,

Particularly, if are WUOD, then for each and any ,

Following the wide dependence structures as above, we will consider a generalized compound renewal risk model satisfying Assumption and the following specific assumptions.

Assumption

The interarrival times are nonnegative, identically distributed and WLOD r.v.s with finite mean .

Assumption

The claim sizes caused by th accident are nonnegative, identically distributed and WUOD r.v.s, and the other statements of Assumption are still valid.

The rest of this work is organized as follows: in Section 2 we will state the motivations and main results of this paper after presenting some existing results, and in Section 3 we will give some lemmas and then prove the main results.

2. Main Results

In this section, we will present our main results of this paper. Before this, we prepare some related results and the motivations of the main results. For later use, we define .

2.1. Related Results and Motivations

As mentioned above, the asymptotics for the finite-time ruin probability in the compound renewal risk model have been studied by many authors. Among them, Aleškevičienė et al. [2] considered the standard compound renewal risk model with condition (1.7) and showed that

(i)if , and for some , then it holds uniformly for all that

(ii)if , and for some , then it holds uniformly for all that

(iii)if , for some constant , and for some , then it holds uniformly for all that

Recently, Zhang et al. [3] extended the results of Aleškevičienė et al. [2] to the case that the claim sizes caused by th accident are negatively associated and obtained a unified form of as follows: let for , and one of the conditions below holds: (i) for , , and for some ; (ii) for , , and for some ; then it holds uniformly for all that

Observe the results of Zhang et al. [3] especially extended case (iii) of Aleškevičienė et al. [2]. Also, Lin and Shen [4] considered a generalized compound renewal risk model with satisfying one type of asymptotically quadrant subindependent structure and also obtained the same relations (2.2), (2.3), and (2.4) as that of Aleškevičienė et al. [2].

Inspired by the above results, we will further discuss some issues as follows:

(1)to cancel the moment condition on , namely, for some , and for some ;(2)to extend partially the class or to the class ;(3)to discuss the case when are WUOD and are WLOD;(4)to drop the interrelationships between and and investigate the case when both and are heavy tailed.

In the paper, we will answer the four issues directly, and then we obtain our main results in the next section.

2.2. Main Results

For the main results of this paper, we now state some conditions which are that of Wang et al. [9].

Condition 1. The interarrival times are NLOD r.v.s.

Condition 2. The interarrival times are WOD r.v.s and there exists a positive and nondecreasing function such that , for some , , and for all , where means that for all and some finite constant .

Condition 3. The interarrival times are WOD r.v.s with for some and there exists a constant such that .

Condition 4. The interarrival times are WOD r.v.s with for some and for any .

The first main result of this paper is the following.

Theorem 2.1. Consider the generalized compound renewal risk model with Assumptions , , and and condition (1.7), there exists a finite constant such that

Meanwhile, let one of Conditions 1–4 hold, and one of Conditions 1–4 with replaced by still holds.

(1)If for some , additionally, for and ; for and ; for and , then relation (2.4) holds uniformly for all .(2)If and , relation (2.4) holds uniformly for all .

Note that there do exist some WLOD r.v.s satisfying condition (2.5), see Wang et al. [19]. In the second main result below, we discuss the random time ruin probability, which requires another assumption.

Assumption

Let be nonnegative r.v. and independent of the sequences , , and .

Define .

Theorem 2.2. Under conditions of Theorem 2.1 and Assumption , one has

(1)if for some , additionally, for and ; for and ; for and , then it holds uniformly for all that

(2)if and , then relation (2.6) still holds uniformly for all .

Remark 2.3. According to the proofs below of Theorems 2.1 and 2.2, we can see that Conditions 1–4 are doing nothing more than making and satisfy the strong law of large number, namely,

So, Conditions 1–4 in Theorems 2.1 and 2.2 can be replaced by (2.7).

3. Proofs of Main Results

In this section we will give the proofs of our main results, for which we need some following lemmas.

Lemma 3.1. If are WUOD and nonnegative r.v.s with distributions , respectively, then for any fixed ,

holds uniformly for all .

Proof. See Lemma 3.1 of Gao et al. [25].

Particularly, let in Lemma 3.2, we have a lemma below.

Lemma 3.2. If are WUOD and nonnegative r.v.s with distributions , then

Lemma 3.3. If are WUOD and real-valued r.v.s with common distribution and mean and satisfying

then for any , there exists a constants such that

holds for all and all .

Proof. By Proposition 1.2 and along the same lines of the proof of Theorem 3.1 of Tang [26] with slight modifications, we can derive that, for some positive integer ,

From Lemma 3.2, we have

Combining (3.5) and (3.6), there exists a constant such that (3.4) holds for all and all .

The following lemma discusses the strong law of large numbers for widely dependent r.v.s, which is due to Wang and Cheng [27].

Lemma 3.4. Let be a sequence of real-valued r.v.s with finite mean and satisfy one of the Conditions 1–4 with replaced by . Then

Proof. Follow Theorem of Matula [28], Theorem and the proofs of Theorems and 1.2 of Wang and Cheng [27], respectively.

The lemma below gives the tail behavior of random sum, which extends the results of Aleškevičienė et al. [2] and Zhang et al. [3].

Lemma 3.5. Let be a sequence of identically distributed and real-valued r.v.s with distribution and finite mean and satisfy one of the Conditions 1–4 with replaced by , where for Conditions 2 and 3 one further assumes that (3.3) holds. Let be nonnegative integer-valued r.v. with distribution and finite mean , independent of . Assume that for some .

(1)If , , and the conditions of Lemma 3.4 are valid, then

(2)If and , then relation (3.8) holds.(3)Let no assumption be made on the interrelationship between and . If , , and the conditions of Lemma 3.4 are still valid, then relation (3.8) still holds.

Proof. Because has finite mean, there exists a large integer such that, for any fixed , it holds that

First consider the case that . Clearly, implies . For any and any , we have

For , by Lemma 3.2 it follows that

For , since and , we obtain by Lemma 3.3 that

where , and are two constants only depending on . Hence, applying (3.9), Lemma 3.2, and the dominated convergence theorem can yield that

For , since can be arbitrarily close to , we see by that

Substituting (3.11)–(3.14) into (3.10) and considering the arbitrariness of , we derive that

On the other hand, we note that

For , by Lemma 3.2 and (3.9), we get

For , by Lemma 3.4 we find that

which, along with and the arbitrariness of , leads to

Hence, from (3.16)–(3.19) and the arbitrariness of , we obtain that

So, combining (3.15) and (3.20) proves that (3.8) holds for .

Next we turn to the case that , namely, . According to Lemma 4.4 of Fa et al. [29], there exists a nondecreasing slowly varying function such that , which results in that for some ,

Define

where . Let

It is easy to verify that are still WUOD and identically distributed r.v.s with common distribution . By the definition of , we know that , , and then . Thus, in (3.10) is divided into three parts as

Clearly, are such that the conditions of Lemmas 3.2 and 3.3 hold, then (3.11) and (3.13) can still hold with , , and replaced by , , and . So, we deduce by and that

For , it follows from Lemma 3.4 that

Then, by we have

From (3.10), (3.14), and (3.24)–(3.27), we find that

Again by (3.16) and (3.19), it is seen that

Since and , we get

Consequently, we obtain by combining (3.28)–(3.30) that (3.8) holds for .

Now we deal with the case that , namely, . Apparently, when , we can derive by Lemmas 3.2 and 3.3 that (3.11) and (3.13) still hold. As for , by and , we know that

Substituting (3.11), (3.13), and (3.31) into (3.10) implies that

For , by Lemma 3.2 we also get (3.17). As for , arguing as (3.19) and (3.31), we still have that

From (3.16), (3.17), and (3.33), we conclude that

Similarly to the derivation of (3.30), by and we still see that . This, along with (3.32) and (3.34), gives relation (3.8) immediately.

According to the proof of (2), we know that if , then (3.11) and (3.13) hold. While from the proof of (1), we have (3.14) if . Hence, under the conditions of (3), we obtain (3.15).

On the other hand, from the proof of (2), we can get (3.17) when . Again by the proof of (1), relation (3.19) also holds for . So, (3.20) is proved under the conditions of (3). As a result, we show (3.8) directly.

The next two lemmas will give some results of the renewal risk model, which is the compound renewal risk model with .

Condition 5. For in (2.5), there exist and in (1.12) such that .

Lemma 3.6 (Corollary 2.1 of Wang et al. [9]). Consider the compound renewal risk model with and , in which are i.i.d. r.v.s with common distribution , and are WLOD r.v.s satisfying (2.5) and one of Conditions 1–4.

(1)If , then for any , it holds uniformly for all that

(2)Furthermore, if Condition 5 holds, then relation (3.35) still holds uniformly for all .

Lemma 3.7 (Corollary of Wang et al. [9]). Under conditions of Lemma 3.6 and assumption , one has

(1)if , then

holds uniformly for all for any .(2)Additionally, if Condition 5 holds, then relation (3.36) still holds uniformly for all .

Now we prove the main results as follows.

Proof of Theorem 2.1. Clearly, if then satisfies Condition 5. In fact, the assumption indicates that the tail of behaves essentially like a power function, thus there exists such that . Take for any , which satisfies (1.12). So, for any , and Condition 5 holds.

First, consider Theorem 2.1(1). For the case that , we can obtain relation (2.4) by (1.5) and Lemmas 3.5 and 3.6, and going along the similar ways to that of Theorems  2.2 and of Aleškevičienė et al. [2]. For the case that , since , by Lemma 3.5 we have that for any ,

which tell us that the distribution of belongs to the class . Then, arguing as the proof of Theorem 2.2 of Aleškevičienė et al. [2], we also obtain relation (2.4).

Now consider Theorem 2.1. By Lemma 3.5,  and , it is also clear that the distribution of belongs to . Hence, relation (2.4) still holds from (1.5) and Lemmas 3.5and 3.6.

Proof of Theorem 2.2. By (1.6), the uniformity of (2.4) in Theorem 2.1, and the independence between and the risk system, we can get the proof of Theorem 2.2.

Acknowledgments

The authors would like to thank Professor Yuebao Wang for his thoughtful comments and also thank the editor and the anonymous referees for their very valuable comments on an earlier version of this paper. The work is supported by Research Start-up Foundation of Nanjing Audit University (no. NSRC10022), Natural Science Foundation of Jiangsu Province of China (no. BK2010480), and Natural Science Foundation of Jiangsu Higher Education Institutions of China (no. 11KJD110002).

References

Q. Tang, C. Su, T. Jiang, and J. Zhang, “Large deviations for heavy-tailed random sums in compound renewal model,” Statistics & Probability Letters, vol. 52, no. 1, pp. 91–100, 2001.

A. Aleškevičienė, R. Leipus, and J. Šiaulys, “Tail behavior of random sums under consistent variation with applications to the compound renewal risk model,” Extremes, vol. 11, no. 3, pp. 261–279, 2008.

J. Zhang, F. Cheng, and Y. Wang, “Tail behavior of random sums of negatively associated increments,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 64–73, 2011.

Z. Lin and X. Shen, “Approximation of the tail probability of dependent random sums under consistent variation and applications,” Methodology and Computing in Applied Probability. In press

Yang, Y. Wang K, and J. Liu, “Asymptotics and uniform asymptotics for finite-time and infinite-time absolute ruin probabilities in a dependent compound renewal risk model,” Journal of Mathematical Analysis and Applications, vol. 398, no. 1, pp. 352–361, 2013.

Q. Tang, “Asymptotics for the finite time ruin probability in the renewal model with consistent variation,” Stochastic Models, vol. 20, no. 3, pp. 281–297, 2004.

R. Leipus and J. Šiaulys, “Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes,” Insurance: Mathematics & Economics, vol. 40, no. 3, pp. 498–508, 2007.

Y. Yang, R. Leipus, J. Šiaulys, and Y. Cang, “Uniform estimates for the finite-time ruin probability in the dependent renewal risk model,” Journal of Mathematical Analysis and Applications, vol. 383, no. 1, pp. 215–225, 2011.

Y. Wang, Z. Cui, K. Wang, and X. Ma, “Uniform asymptotics of the finite-time ruin probability for all times,” Journal of Mathematical Analysis and Applications, vol. 390, no. 1, pp. 208–223, 2012.

V. P. Chistyakov, “A theorem on sums of independent positive random variables and its applications to branching processes,” Theory of Probability and Its Applications, vol. 9, pp. 640–648, 1964.

D. A. Korshunov, “Large deviation probabilities for the maxima of sums of independent summands with a negative mean and a subexponential distribution,” Theory of Probability and Its Applications, vol. 46, no. 2, pp. 355–366, 2001.

W. Feller, “One-sided analogues of Karamata's regular variation,” L'Enseignement Mathématique, vol. 15, pp. 107–121, 1969.

D. B. H. Cline, “Intermediate regular and variation,” Proceedings of the London Mathematical Society, vol. 68, no. 3, pp. 594–616, 1994.

D. B. H. Cline and G. Samorodnitsky, “Subexponentiality of the product of independent random variables,” Stochastic Processes and their Applications, vol. 49, no. 1, pp. 75–98, 1994.

C. Klüppelberg, “Subexponential distributions and integrated tails,” Journal of Applied Probability, vol. 25, no. 1, pp. 132–141, 1988.

P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance, vol. 33 of Applications of Mathematics, Springer, Berlin, Germany, 1997.

D. Denisov, S. Foss, and D. Korshunov, “Tail asymptotics for the supremum of a random walk when the mean is not finite,” Queueing Systems, vol. 46, no. 1-2, pp. 15–33, 2004.

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, vol. 27 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1987.

K. Wang, Y. Wang, and Q. Gao, “Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate,” Methodology and Computing in Applied Probability. In press

N. Ebrahimi and M. Ghosh, “Multivariate negative dependence,” Communications in Statistics A, vol. 10, no. 4, pp. 307–337, 1981.

H. W. Block, T. H. Savits, and M. Shaked, “Some concepts of negative dependence,” The Annals of Probability, vol. 10, no. 3, pp. 765–772, 1982.

L. Liu, “Precise large deviations for dependent random variables with heavy tails,” Statistics & Probability Letters, vol. 79, no. 9, pp. 1290–1298, 2009.

Y. Chen, K. C. Yuen, and K. W. Ng, “Precise large deviations of random sums in presence of negative dependence and consistent variation,” Methodology and Computing in Applied Probability, vol. 13, no. 4, pp. 821–833, 2011.

K. Joag-Dev and F. Proschan, “Negative association of random variables, with applications,” The Annals of Statistics, vol. 11, no. 1, pp. 286–295, 1983.

Q. Gao, P. Gu, and N. Jin, “Asymptotic behavior of the finite-time ruin probability with constant interest force and WUOD heavy-tailed claims,” Asia-Pacific Journal of Risk and Insurance, vol. 6, no. 1, 2012.

Q. Tang, “Insensitivity to negative dependence of the asymptotic behavior of precise large deviations,” Electronic Journal of Probability, vol. 11, no. 4, pp. 107–120, 2006.

Y. Wang and D. Cheng, “Basic renewal theorems for random walks with widely dependent increments,” Journal of Mathematical Analysis and Applications, vol. 384, no. 2, pp. 597–606, 2011.

P. Matuła, “A note on the almost sure convergence of sums of negatively dependent random variables,” Statistics & Probability Letters, vol. 15, no. 3, pp. 209–213, 1992.

G. Faÿ, B. González-Arévalo, T. Mikosch, and G. Samorodnitsky, “Modeling teletraffic arrivals by a Poisson cluster process,” Queueing Systems. Theory and Applications, vol. 54, no. 2, pp. 121–140, 2006.