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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.

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.

It is well known that the compound renewal risk model was first introduced by Tang et al. [

The interarrival times

The claim sizes and their number caused by

The sequences

Denote the arrival times of the

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

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

Chistyakov [

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. [

Say that r.v.s

The WUOD, WLOD, and WOD r.v.s are collectively called as widely dependent r.v.s. Recall that if

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

Particularly, if

Following the wide dependence structures as above, we will consider a generalized compound renewal risk model satisfying

The interarrival times

The claim sizes caused by

The rest of this work is organized as follows: in Section

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

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. [

if

if

if

Recently, Zhang et al. [

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

to cancel the moment condition on

to extend partially the class

to discuss the case when

to drop the interrelationships between

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

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

The interarrival times

The interarrival times

The interarrival times

The interarrival times

The first main result of this paper is the following.

Consider the generalized compound renewal risk model with Assumptions

If

If

Note that there do exist some WLOD r.v.s satisfying condition (

Let

Define

Under conditions of Theorem

if

if

According to the proofs below of Theorems

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

If

See Lemma

Particularly, let

If

If

By Proposition

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

Let

Follow Theorem

The lemma below gives the tail behavior of random sum, which extends the results of Aleškevičienė et al. [

Let

If

If

Let no assumption be made on the interrelationship between

Because

Next we turn to the case that

On the other hand, from the proof of (2), we can get (

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

For

Consider the compound renewal risk model with

If

Furthermore, if Condition

Under conditions of Lemma

if

Additionally, if Condition

Now we prove the main results as follows.

Clearly, if

First, consider Theorem

Now consider Theorem

By (

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).