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ISRN Applied Mathematics
Volume 2013 (2013), Article ID 936301, 10 pages
Precise Large Deviations for Random Sums of END Random Variables with Dominated Variation
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China
Received 26 February 2013; Accepted 8 May 2013
Academic Editors: E. Di Nardo and D. Luchinsky
Copyright © 2013 Yu Chen and Zhihui Qu. 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.
We investigate the precise large deviations for random sums of extended negatively dependent random variables with long and dominatedly varying tails. We find out that the asymptotic behavior of precise large deviations of random sums is insensitive to the extended negative dependence. We apply the results to a generalized dependent compound renewal risk model including premium process and claim process and obtain the asymptotic behavior of the tail probabilities of the claim surplus process.
The study of large deviations plays an important role in insurance and finance theory. The aim of this paper is to study precise large deviation probabilities for sequences of dependent and heavy-tailed random variables. Let be a sequence of real-valued heavy-tailed random variables with common distribution function and finite mean . We denote its tail by . We say that (or its distribution ) is heavy-tailed if it has no exponential moments. Suppose that is a nonnegative integer-valued process independent of the sequence with mean function which tends to as . For random sums of the type we aim to study the precise large deviation probabilities
Some earlier work, for the case are independent, we refer the reader to see Cline and Hsing , Klüppelberg and Mikosch , Tang et al. , Ng et al. , Konstantinides and Loukissas  and Loukissas , and so forth; for the case are negatively dependent, see Chen and Zhang , Tang , Konstantinides and Loukissas , Chen et al.  and Wang et al. , and so forth.
Definition 1. We call random variables extended negatively dependent (END), if there exists a constant such that hold for each and all .
Recall that if for any integer in (3) and (4), then random variables are called lower negatively dependent (LND) and upper negatively dependent (UND), respectively. are called negatively dependent (ND) if both (3) and (4) hold, so an ND sequence must be an END sequence. From Liu , the END structure can reflect a negative dependence structure but also a positive one to some extent, so the END structure is substantially more comprehensive than the ND structure.
In presence of END structure, Liu  obtained that the precise large deviations of partial sums with consistent variation are insensitive to the dependence structure. Chen et al.  extended the previous results of Liu  to random sums with consistently varying tails. Wang et al.  considered a wider dependent structure and investigated the precise large deviations for the partial sums with dominatedly varying tails.
The basic assumption of this paper is that is a sequence of real-valued END random variables with common heavy-tailed distribution and finite mean . We consider a wider heavy-tailed distribution class than consistently varying class, the intersection of long-tailed class, and dominatedly varying-tailed class. The main aim of this paper is to extend the study to certain END cases and find out whether the asymptotic behavior of precise large deviations is insensitive to the extended negative dependence.
The rest of the paper is organized as follows. After simply reviewing some subclasses of heavy-tailed distributions and giving some lemmas needed to prove the theorem in Section 2, we derive our main result of precise large deviations for random sums in Section 3 and apply the result to the dependent compound renewal risk model in Section 4.
2.1. Heavy-Tailed Distribution Classes
To formulate our main results we need to introduce some notations and assumptions. Throughout this paper we restrict ourselves to the case that is a sequence of heavy-tailed random variables. In risk theory, heavy-tailed distributions are often used to model large claim amounts. They play a key role in some fields such as insurance, financial mathematics, and queueing theory. Next, we recall some important subclasses of heavy-tailed distributions.
A distribution function (d.f.) is said to belong to class , if holds for some ; denote that . Such a d.f. is said to have a dominatedly varying tail.
A d.f. is said to belong to class , if holds for all ; denote that . Such a d.f. is said to have a long tail.
A d.f. is said to belong to class , if Denote that . Now the d.f. is said to have a consistently varying tail.
It is well known that the previous heavy-tailed distribution classes have the following relationship:
For a d.f. , we define and ; we call the (upper) Matuszewska index of the d.f. . From Bingham et al. [15, Chapter 2.1], we know that the following assertions are equivalent: From the definition of the class , it holds that if and only if . Denote that If , we know that .
We close this section by explaining some symbols which will be used later. We will use , and to connect two positive functions, say and , as follows: We say that and are weakly equivalent if . Throughout, every limit relation without explicit limit is with respect to .
2.2. Some Lemmas
We will need some lemmas used in the proofs of our theorems. The following lemma is given by Tang .
Lemma 2. If , then(1)for each , there exist positive constants C and D such that the inequality holds for all ;(2)it holds for each that .
Lemma 3. If , then for any fixed satisfying , holds uniformly for .
Proof. The proof is analogous to that of Lemma 2.2 of Chen and Zhang  with some minor modifications and is omitted.
By Definition 1, the following properties of END sequences can be obtained directly.
Lemma 4. Let be a sequence of END random variables, and then(1) are still END, where are either all monotone increasing or all monotone decreasing;(2)for any , there exists a constant such that
Wang et al.  proved the following result for partial sums.
Lemma 5. Let be a sequence of END random variables with common distribution and finite mean , satisfying Then, for any ,
Lemma 6. Let be a sequence of END random variables with common distribution and finite mean , satisfying Then, for any ,
Proof. Let , and denote its distribution by . It is easy to see that and
(i) Consider .
By and , it is easy to see that . In fact for any , By the definition , we see that Thus, by Lemma 5, we have
(ii) Consider .
First we shows that . For large there exist such that . For some and , By the definition , we see that Similarly, by Lemma 5, we have
Lemma 7. Let be END with common distribution and mean , satisfying (18). Denote that . Then, for each fixed and some irrespective to and , the inequality holds uniformly for all and .
Proof. Firstly, by Lemma 5, we know that holds uniformly for all . Moreover, for each fixed positive integer , by Lemma 2, the inequality holds for some , all , and all . Hence, we complete the proof.
Lemma 8. Let be a sequence of END random variables with common distribution and finite mean ; if , then for any and some , the inequality holds for all and .
3. Precise Large Deviations for Random Sums
3.1. Main Theorem
Theorem 9. Let be a sequence of real-valued random variables with common distribution and finite mean , satisfying (18). Let be a nonnegative integer-valued process independent of , and assume that satisfies for some and for all . is defined by (1), and then for any , holds uniformly for .
Remark 10. From the following proof, it is easy to see that (33) holds uniformly for and if just satisfies Assumption .
Corollary 11. Under the assumptions of Theorem 9 and if , then for any , holds uniformly for .
3.2. Proof of Theorem 9
In the sequel, always represents an absolute positive constant, which may vary from different places.
In order to prove Theorem 9, we divide the random sum into three parts as follows: where is an arbitrarily fixed number to be specified later.
By Tang et al. , Assumption implies that then by the dominated convergence theorem, Therefore, for any ,
Therefore, Theorem 9 originates from the following three lemmas.
Lemma 12. Assume that Assumptions and hold; then for any , holds uniformly for .
Proof. In order to prove this lemma, we consider the following three cases.
(i) and .
Choose such that . Note that , . By Lemma 7, and the last equality holds by relation (39).
(ii) and .
Choose such that . Note that . Mimicking the proof of (i) and by Lemma 7, we have where in the last step, we used the relation .
(iii) and .
Let . For , we have . By Lemma 7, where the last step can be verified as .
For , note that . Hence, by Assumption , the inequality still holds uniformly for all . As a result, the relation holds uniformly for all .
Lemma 13. Assume that Assumptions and hold; then for any , holds uniformly for .
Proof. Similar to the proof of Lemma 12, we split into three cases.
(i) and .
Firstly, for , then . For , By Lemma 6, for any and sufficient large , Then, thus by , letting , we have . Combining this and (37), On the other hand, by the similar way, for , then , We have Combining (50) and (52), we complete the proof of the lemma in this case.
(ii) and .
Similarly, for any and sufficient , Analogously, we can obtain
(iii) and .
Since , note that , and it follows from Lemma 6 that for all , On the other hand, for , Similarly, both (50) and (52) hold in this case.
Lemma 14. Assume that Assumptions and hold, and then for any , holds uniformly for .
Proof. We prove this lemma by splitting into three cases like the former lemmas’ proof.
(i) and .
For , by Lemma 8, there exists a constant such that where in the last step we have used Lemma 2, Assumption and relation (38).
(ii) and .
Similarly, by Lemma 8, there exists a constant such that
(iii) and .
Note that . Therefore, Lemma 6 implies that
4. Applications for Random Sums for a Dependent Compound Renewal Risk Model
4.1. The Dependent Compound Renewal Risk Model
From a realistic point of view, we further generalize such work to a much more realistic model including premium income process. The premium income process depends not only on the number of customers who buy the insurance portfolios but also on the premium size process. The risk model has the following structure.(1)The individual claim sizes are END random variables with common distribution and finite mean .(2)The number of claims in the interval is denoted by . Suppose that is a nonnegative integer-valued process with mean function which tends to as and independent of .(3)The number of customers who buy the insurance portfolios within the time interval is denoted by . Assume that is a strictly stationary renewal counting process with mean function .(4)The premium size process is a sequence of nonnegative UND random variables.
Suppose that is the initial reserve of an insurance company. Thus, the risk reserve process is given by while the claim surplus process is In addition, the random sequences , and are mutually independent. This model (62) is called generalized dependent compound renewal risk model. For this model (62), the net profit condition is that .
Theorem 15. For the generalized dependent compound renewal risk model (62), let be a sequence of random variables with common distribution and finite mean , satisfying (18), and let be a sequence of nonnegative random variables. Assume that satisfies Assumptions and , and then for any , holds uniformly for .
Corollary 16. Under the conditions of Theorem 15, further assume that is a Poisson process with intensity , and then for any with , holds uniformly for .
Remark 17. If we assume that in Theorem 15, we have that holds uniformly for .
From the previous two theorems, we find out that the asymptotic behavior of precise large deviations of random sums is insensitive to the extended negative dependence.
4.2. Proof of Theorem 15
Observing that is a renewal counting process, . According to the strong law of large numbers for UND random variables (see Matuła [17, Theorem 1]), when , we have Consequently, there is a positive function such that , and For convenience, we denote that . For the model (62), we have In order to prove the theorem, we divide this integral into three parts similarly as follows:
We proceed a series of lemmas to prove Theorem 15.
Lemma 18. For in (67), one has for that holds uniformly for .
Lemma 19. For in (67), one has for that holds uniformly for .
Proof. For any and , noting that as , we have for sufficient large ,
Again using Theorem 9, we conclude that holds uniformly for . Moreover, , and then as ; by Lemma 3, we have that holds uniformly for .
Hence, combining (67), (74), and (75), we see that symmetrically, Then the proof of Lemma 19 is finished.
Lemma 20. For in(67), one has for that holds uniformly for .
The first author is supported by National Science Foundation of China (nos. 10801124 and 11171321) and the Fundamental Research Funds for the Central Universities (no. WK 2040170006).
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