上尾渐近独立重尾索赔风险模型的有限时间破产概率的一致渐近性

马皓杰，王开永（苏州科技大学 数学科学学院，江苏 苏州 215009）

1 Introduction

We consider the finite-time ruin probability of a nonstandard risk model，in which the claim sizes{Xi，i≥1}form a sequence of nonnegative，but not identically distributed random variables（r.v.s），and the inter-arrival times{θi，i≥1}，form another sequence of nonnegative r.v.s.The arrival times of successive claimsconstitute a general counting process{N（t），t≥0}.The process{N（t），t≥0}is independent of{Xi，i≥1}and satisfies EN（t）＜∞for all 0＜t＜∞.The total amount of premiums accumulated before time t≥0，denoted by C（t），is a nonnegative and nondecreasing stochastic process with C（0）=0 and C（t）＜∞almost surely（a.s.）for every 0＜t＜∞.We suppose that{C（t），t≥0}，{Xi，i≥1}and{N（t），t≥0}are independent.Let Λ={t:EN（t）＞0}={t:P（τ1≤t）＞0}.Let r≥0 be a constant interest rate and x≥0 be an initial reserve of an insurance company.Denoted the total reserve up to time t≥0 of the insurance company by Ur（t），

where 1Ais the indicator function of the set A.The discounted aggregate claims up to time t≥0 is denoted by

and the discounted value of premiums accumulated before time 0≤t＜∞is denoted by

Then the ruin probability within a finite time t>0 is defined as

In this paper，without special statement，all limit relationships hold for x→∞.For two positive functions a（x）and b（x），we write a（x）～b（x）if lim a（x）/b（x）=1；write a（x）≲b（x）or b（x）≳a（x）if lim sup a（x）/b（x）≤1；writeif 0＜lim inf a（x）/b（x）≤lim sup a（x）/b（x）＜∞，and write a（x）=o（b（x））if lim a（x）/b（x）=0.Furthermore，for two positive bivariate functions a（x，t）and b（x，t），we write a（x，t）～b（x，t）uniformly for all t in a nonempty setΔ，if write a（x，t）≲b（x，t）or b（x，t）≳a（x，t）uniformly for all t∈Δ，if

The dominated variation distribution class is an important class of heavy-tailed distributions，which is denoted by D.A distribution V on（-∞，∞）belongs to the class D if for any y∈（0，1）

For more details about heavy-tailed distributions in the insurance and finance context or literature，the readers can refer to Embrechts et al[1]，Foss et al[2]and so on.For a distribution V on（-∞，∞）and any y＞1，we define

（i）V∈D；（ii）0＜LV≤1；（iii）JV+＜∞.

In this paper，we consider a risk model with dependent claim sizes.In the following，a dependence structure is given.

Definition 1Say that the real-valued r.v.s{ξi，i≥1}are upper tail asymptotically independent（UTAI），if

and

If we change the above relation into

In the following，we introduce some related heavy-tailed distribution classes.For a proper distribution V on（-∞，∞），let=1-V be its（right）tail.A distribution V on（-∞，∞）is said to be heavy-tailed ifthen{ξi，i≥1}are said to be tail asymptotically independent（TAI）.

The UTAI structure was proposed by Geluk and Tang[4]when they investigated the asymptotics of the tail of sums with dependent increments.There are some papers investigating the UTAI structure，such as Liu and Gao[5]，Asimit et al[6]and so on.

It is well know that in the renewal risk model，when the claim sizes and the inter-arrival times are all independent and identically（i.i.d）r.v.s，the ruin probabilities have been widely studied，see Tang[7-8]，Hao and Tang[9].Recently，researchers have been interested in the dependence structures of the claim sizes and the inter-arrival times，see Yang and Wang[10]，Wang et al[11]，Chen and Wang[12]，Yang[13]，and so on.

Liu and Gao[5]consider a risk model，in which the claim sizes form a sequence of upper tail asymptotically independent and identically distributed random variables.Gao and Liu[14]also introduced a risk model，but the claim sizes are non-identically distributed random variables，and the distributions of the claim sizes belong to the intersection of the long-tailed distribution class and the dominated variation distribution class.

The main purpose of this paper is to discuss the uniform asymptotics for the finite-time ruin probability of a risk model with UTAI claim sizes belonging to the class D.Here are the main results of this paper.

Theorem 1For the above risk model，if the claim sizes{Xi，i≥1}are UTAI r.v.s with distributions Fi，i≥1，respectively.Assume that there exists a distribution F∈D such that

Suppose that for any fixed t＞0，the general claim-arrival process{N（t），t≥0}satisfies E（N（t））p+1<∞for some p＞JF+.Then for any fixed T∈Λ，it holds uniformly for all t∈ΛT≡Λ∩[0，T]that

Theorem 2Under the conditions of Theorem 1，for any fixed T∈Λ，it holds uniformly for all t∈ΛTthat

We will prove the main results in Section 3.We first give some lemmas in Section 2.

2 Some lemmas

This section presents some lemmas，which are useful in proving the main results of this paper.The following lemma can be obtained from Proposition 2.2.1 of Bingham et al[3]and Lemma 3.5 of Tang and Tsitsiashvili[15].

Lemma 1Let V be a distribution on（-∞，∞）belonging to the class D.

（1）For any 0

and for all x≥y≥D2，

（2）For any p＞JV+，it holds that x-p=o（Vˉ（x））.

For the second lemma，we can get it directly from Theorem 3.3（iv）of Cline and Samorodnitsky[16]and Lemma 2.5 of Wang et al[17].

Lemma 2Let ξ be a real-valued r.v.with a distribution V∈D，and η be a nonnegative r.v.，which is independent of ξ.If Eηp<∞for some p＞JV+，then

Referring to Lemma 2.4 of Wang et al[18]and Assumption 1.3 of Wang and Chen[19]，we can obtain the following lemma.

Lemma 3For n real-valued numbers ci，1≤i≤n，let cˉn=（c1，…，cn）.Let{ξi，1≤i≤n}be n TAI and real valued r.v.s with distributions Vi，1≤i≤n，respectively.Assume that there exists a distribution V∈D such that

Then for any fixed constant b＞0，

and

ProofBy（10）and V∈D we get that Vi∈D，1≤i≤n，and for any y＞1，i≥1，

Follow the proof of Lemma 2.4 of Wang et al[18]with some slight changes.It follows form Lemma 2.3 of Wang et al[18]that for any ε＞0 there exists y0＞0 such that for any 1≤j≤n，when xi＞y0and xj＞y0

For any x＞0 and any fixed constant L1＞by0，

By（2.7）-（2.9）of Lemma 2.4 of Wang et al[18]，it holds uniformly for cˉn∈（0，b]nthat

Since Vi∈D，1≤i≤n，for any γ1＞1，when x is sufficiently large，it holds uniformly for cˉn∈（0，b]nthat

By（13），it holds uniformly for∈（0，b]nthat

Therefore，

It follows from（14）-（16）that（11）holds.

In the following，we prove（12）.It follows form（5）that for any ε＞0 there exists y1＞0 such that for any 1≤i≠j≤n，when xi＞y1and xj＞y1

For any x＞0 and any fixed constant L2＞bny1，

For J1，since Vi∈D，1≤i≤n，for any 0＜γ2＜1，when x is sufficiently large，it holds uniformly for cˉn∈（0，b]nthat

Hence，by（13），it holds uniformly for∈（0，b]nthat

Then

For J2，by the proof of Lemma 2.4 of Wang et al[18]it holds that

By（17）-（19）we get that（12）holds.□

The following lemma is Lemma 3.4 of Gao and Liu[14].

Lemma 4Under the conditions of Theorem 1，for any fixed t∈Λ，

3 Proofs of main results

Now we begin to prove the main results of this paper.First，we should give the proof of Theorem 2 which is helpful for proving Theorem 1.

3.1 Proof of Theorem 2

We follow the line of the proof of Theorem 2.4 of Gao and Liu[14].For an arbitrarily fixed integer m，we have

By the proof of Theorem 2.4 of Gao and Liu[14]and Lemma 2，we can obtain that uniformly for all t∈ΛT.

For k2，it holds uniformly for all t∈ΛTand 1≤n≤m that

where H（t1，…，tn+1）denotes the joint distribution of（τ1，…，τn+1），1≤n≤m.

By Lemma 3，it holds uniformly for all t∈ΛTand 1≤n≤m that

Thus，it holds uniformly for all t∈ΛTand 1≤n≤m that

Therefore，we have it holds uniformly for all t∈ΛT.

By（4.7）and（4.8）of Gao and Liu[14]，it holds uniformly for all t∈ΛTthat

which combining with（20）-（22）and Lemma 4 yields that（8）holds uniformly for all t∈ΛT.□

3.2 Proof of Theorem 1

From（1）-（4），we have for any x＞0 and t＞0，

Hence，for any x＞0 and t∈ΛTit holds that

and

From（23）and Theorem 2，it holds uniformly for all t∈ΛTthat

By（24），and Theorem 2，for any q＞0，it holds uniformly for all t∈ΛTthat

By（26），F∈D and Fatou’s lemma，we have for any δ＞1，when x is sufficiently large

Letting δ↘1 and q↗∞，it holds uniformly for all t∈ΛT

By（25）and（27）we get that（7）holds.□