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The study of precise large deviations of random sums is an important topic in insurance and finance. In this paper, extended precise large deviations of random sums in the presence of END structure and consistent variation are investigated. The obtained results extend those of Chen and Zhang (2007) and Chen et al. (2011). As an application, precise large deviations of the prospective- loss process of a quasirenewal model are considered.

In the risk theory, heavy-tailed distributions are often used to model large claims. They play a key role in some fields such as insurance, financial mathematics, and queueing theory. We say that a distribution function

Throughout this paper, let

Recently, for practical reasons, precise large deviations of random sums with heavy tails have received a remarkable amount of attention. The study of precise large deviations is mainly to describe the deviations of a random sequence or a stochastic process away from its mean. The mainstream research of precise large deviations of

In this paper, we are interested in the deviations of random sums

The basic assumption of this paper is that

One calls random variables

Recall that

Under the assumption that

The rest of this paper is organized as follows. Section

Throughout this paper, by convention, we denote

Next we will need some lemmas in the proof of our theorems. From Lemma 2.3 of Chen et al. [

Let

Lemma

Let

In this sequel, all limiting relationships, unless otherwise stated, are according to

For any

The relation

One can easily see that Assumption

Let

Let

Assume that

Assume that

One can easily see that Theorem

Under the conditions of Theorem

We use the commonly used method with some modifications to prove Theorem

We first deal with

Next, for

Finally, to deal with

Substituting (

(i) We also start with the decomposition (

For

Again, as

Finally, in

(ii) We also start with the representation (

To deal with

For

Finally, in

Substituting (

In this section we consider precise large deviations of the prospective-loss process of a quasirenewal model, where the quasi-renewal model was first introduced by Chen et al. [

Let

The authors would like to thank an anonymous referee for his/her constructive and insightful comments and suggestions that greatly improved the paper. This work was partially supported by NSFC Grant 11071076, the Talents Youth Fund of Anhui Province Universities (2011SQRL012ZD), the Project Sponsored by the Doctoral Scientific Research Foundation of Anhui University, and the 211 Project of Anhui University (2009QN020B).