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）

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.

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，

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.□