Precise large deviations of aggregate claims in a size-dependent renewal risk model with stopping time claim-number process

In this paper, we consider a size-dependent renewal risk model with stopping time claim-number process. In this model, we do not make any assumption on the dependence structure of claim sizes and inter-arrival times. We study large deviations of the aggregate amount of claims. For the subexponential heavy-tailed case, we obtain a precise large-deviation formula; our method substantially relies on a martingale for the structure of our models.


Introduction
Consider the following renewal risk model. Let {X k , k ∈ N} and {θ k , k ∈ N} be claim sizes and inter-arrival times, respectively. Assume that (X k , θ k ), k ∈ N, form a sequence of independent and identically distributed (i.i.d.) copies of a generic random pair (X, θ ) with marginal distribution functions F =  -F on [, ∞) and G on [, ∞), with dependent components X and θ . The claim arrival times are τ k = k i= θ i , k ∈ N, with τ  = . The number of claims is defined by then N * t is a stopping time. In this way, the aggregate amount of claims over the [, t] is of the form We study large deviations of S * t in (.). We only consider the case of heavy-tailed claimsize distributions. One of the most important classes of heavy-tailed distributions is the holds for all n ≥ , where F * n denotes the n-fold convolution of F. Clearly, (.) implies A non-standard renewal risk model with dependent components X and θ , which was firstly proposed by Albrecher and Teugels [] and further studied by Boudreault et al.
holds for t ≥  and large enough x, they studied the large deviations of the aggregate amount of C heavy-tailed claims, where C ⊂ S (see Embrechts et al. []). We now comment on the approaches used in this work. First, in Theorem ., we do not make an assumption on the dependence structure of (X, θ ). The existing results usually require a conditional tail probability of X given θ , e.g., Chen and Yuen [] made the assumption (.), to say the least. Second, we extend the asymptotic behavior of the large deviations of S * t to the case of S heavy-tailed claims. Finally, we construct a martingale to prove our result.
The rest of the paper is organized as follows. Section  recalls various preliminaries and prepares a few lemmas. Section  presents the proof of the main result. We end the paper with conclusions in Section .

Preliminaries
Throughout this paper, for two positive functions a(·) and b(·), we write a( if both. 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(·, ·) ∼ b(·, ·) holds uniformly for Clearly, the asymptotic relation a(·, ·) ∼ b(·, ·) holds uniformly for x ∈ if and only if To obtain our desired results, we need to mention the following useful lemma.
Lemma . Consider the renewal counting process N * t in (.). Under the assumption E[θ ] = /λ < ∞. Then, for any p ≥ , we have Proof First, for arbitrarily ε > , by definition of N * and E[θ ] = /λ, we have Combining (.) and (.), we have Indeed, for any δ > , For the case when u ≥  is an integer By the law of large numbers So we have the bound, for any constant c > , Combining (.) and (.), uniformly for large u and t, By (.) and (.), we obtain Lemma ..

Proof of Theorem 1.1
By F ∈ S and (.), we need only to prove P max Write ξ k = I {X k -μ>x} . First, By Wald's equation On the other hand, All we need is to bound the second term. Notice that M n ≡ ≤j<k≤n ξ j (ξ k -Eξ k ), n = , , . . . is a martingale. By Doob's stopping rule, Hence, Write Z n = n j= ξ j . Notice that Therefore,