Large deviations for random sums of differences between two sequences of random variables with applications to risk theory
© Yang et al.; licensee Springer 2012
Received: 14 February 2012
Accepted: 11 October 2012
Published: 29 October 2012
This paper investigates some precise large deviations for the random sums of the differences between two sequences of independent and identically distributed random variables, where the minuend random variables have subexponential tails, and the subtrahend random variables have finite second moments. As applications to risk theory, the customer-arrival-based insurance risk model is considered, and some uniform asymptotics for the ruin probabilities of an insurance company are derived as the number of customers or the time tends to infinity.
MSC:60F10, 62E20, 62P05.
1 Introduction and main result
for some . Clearly, the condition (1.1) implies (1.2); see, e.g., Tang et al. .
Asymptotic formulae that hold with such a uniformity feature are usually of higher theoretical and practical interests. The indicator function of an event A is denoted by .
To formulate our main results, we firstly introduce some notations and assumptions. Let , , be the hazard function of the distribution B. We assume that there exists a nonnegative function q such that , , which is called the hazard rate of B. Denote the hazard ratio index by . The following condition is essential for our purposes.
for some and any . Comparing with this condition, (1.2) is weaker due to Lemma 2.5 of Ng et al. . The condition (1.3) is also satisfied for, e.g., the renewal counting process generated by independent or some dependent r.v.s according to some elementary renewal theorems (see, e.g., Proposition 2.5.12 of Embrechts et al. , Theorem 6.1 of Yang and Wang , Theorems 1.2 and 1.3 of Wang and Cheng etc.).
Throughout the paper, we assume that . Under Condition A, we state our main results below.
holds uniformly for all .
Corollary 1.1 Assume that Y has a finite second moment, (1.2) and (1.3) hold. If the hazard function of r.v. is of the form , where , then for any , (1.4) holds uniformly for all .
For some applications of these results in insurance, finance and queueing system, one can refer to Klüppelberg and Mikosch , Mikosch and Nagaev , Baltrūnas et al.  among others. In Section 2 we consider the customer-arrival-based insurance risk model (CIRM) and obtain some uniformly asymptotic behavior of the accumulated risks of an insurance company as the number of customers tends to infinity and the time tends to infinity. The proofs of Theorem 1.1 and Corollary 1.1 will be postponed in Section 3.
2 Applications to risk theory
The customer-arrival process is a general counting process, namely a nonnegative, nondecreasing, right continuous and integer-valued random process. Denote the times of successive customer-arrival by , .
At the time , the n th customer purchases an insurance policy. Assume that an insurance period lasts . Then in an insurance period , the insurance company has a potential risk of payment.
The potential claims , independent of , are nonnegative i.i.d. r.v.s with a common distribution B and a finite mean . The price of an insurance policy is , where the positive constant ρ is interpreted as a relative safety loading. The net loss of the n th customer is .
This model was introduced by Ng et al. . Clearly, Lemma 3.1 and Theorem 1.1 lead to some precise large deviation results for the processes and in the CIRM.
- (i)Assume that Condition A holds, then for any(2.1)
- (ii)Assume that Condition A, (1.2) and (1.3) hold. If , then for any(2.2)
We address that the large deviation problems for the prospective loss process describe the uniformly asymptotic behavior of the accumulated risks.
3 Proof of main result
In the sequel, the constant C always represents a positive constant, which may vary from place to place. Before proving Theorem 1.1, we require some lemmas.
decreases for sufficiently large u;
for any , there exist positive and such that for .
where is some positive function satisfying and . We proceed with a series of lemmas below to prove Theorem 1.1.
holds uniformly for all .
holds uniformly for all . It completes the proof of the lemma. □
holds uniformly for all .
holds uniformly for all . □
holds uniformly for all .
holds uniformly for all . Therefore, the desired (3.13) follows from (3.15)-(3.17). □
Combining above Lemmas 3.3, 3.4 and 3.5, we complete the proof of Theorem 1.1.
which implies . □
This paper was supported by the National Natural Science Foundation of China (No. 11001052, 11101394), National Science Foundation of Jiangsu Province of China (No. BK2010480), Qing Lan Project, the Project of Construction for Superior Subjects of Audit Science & Technology/Statistics of Jiangsu Higher Education Institutions.
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