Uniform Estimate of the Finite-time Ruin Probability for All times in a Generalized Compound Renewal Risk Model

Copyright q 2012 Qingwu Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.


Introduction
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.

Risk Model
It is well known that the compound renewal risk model was first introduced by Tang et

Advances in Decision Sciences
Assumption H 1 The interarrival times {θ i , i ≥ 1} of successive accidents are nonnegative, identically distributed, but not necessarily independent r.v.s with finite mean λ −1 .
Assumption H 2 The where x ≥ 0 is the initial surplus and c > 0 is the constant premium rate.The finite-time ruin probability within time t > 0 is defined as Clearly, the ruin can only arise at the times τ n , 1 ≤ n ≤ N t , then Let τ be a nonnegative r.v., the random time ruin probability is In order for the ultimate ruin not to be certain, we assume the safety loading condition holds, namely, 1.7 In the generalized compound renewal risk model above, if all the sequences {θ i , i ≥ 1},

Heavy-Tailed Distribution Classes
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 x → ∞ unless mentioned otherwise.
For a distribution V on −∞, ∞ , denote its tail by V x 1 − V x , and its upper and lower Matuszewska indices by, respectively, for y > 1, where V * y lim inf V xy /V x and V * y lim sup V xy /V x .Chistyakov 10 introduced an important class of heavy-tailed distributions, the subexponential class.By definition, a distribution V on 0, ∞ belongs to the subexponential class, denoted by V ∈ S, if where V * 2 denotes the 2-fold convolution of V .Clearly, if V ∈ S then V is long tailed, denoted by V ∈ L and characterized by One can easily see that a distribution V ∈ L if and only if there exists a function f

Wide Dependence Structure
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. 19 as follows.
Definition 1.1.Say that r.v.s {ξ i , i ≥ 1} are widely upper orthant dependent WUOD , if for each n ≥ 1, there exists some finite positive number g U n such that, for all Say that r.v.s {ξ i , i ≥ 1} are widely lower orthant dependent WLOD , if for each n ≥ 1, there exists some finite positive number g L n such that, for all Furthermore, {ξ i , i ≥ 1} are said to be widely orthant dependent WOD if they are both WUOD and WLOD.
The WUOD, WLOD, and WOD r.v.s are collectively called as widely dependent r.v.s.Recall that if g U n ≡ 1 or g L n ≡ 1 for each n ≥ 1 in Definition 1.1, then {X i , i ≥ 1} are negatively upper orthant dependent or negatively lower orthant dependent NLOD , see Ebrahimi and Ghosh 20 or Block et al.21 ; if g U n g L n ≡ M for some constant M > 0 and each n ≥ 1 such that the two inequalities in Definition 1.1 both hold, then {X i , i ≥ 1} are extended negatively dependent, see Liu 22 and Chen et al. 23 .Obviously, the WUOD and WLOD structures allow a wide range of negative dependence structures among r.v.s, such as extended negative dependence, negatively upper orthant dependence/negatively lower orthant dependence, negative association see Joag-Dev and Proschan 24 , and even some positive dependence.For some examples to illustrate that the WUOD and WLOD structures allow some negatively and positively dependent r.v.s, we refer the readers to Wang et al. 19 .
The following properties for widely dependent r.v.s can be obtained immediately below.
2 If {ξ i , i ≥ 1} are nonnegative and WUOD, then for each n ≥ 1, Particularly, if {ξ i , i ≥ 1} are WUOD, then for each n ≥ 1 and any s > 0, Following the wide dependence structures as above, we will consider a generalized compound renewal risk model satisfying Assumption H 3 and the following specific assumptions.The rest of this work is organized as follows: in Section 2 we will state the motivations and main results of this paper after presenting some existing results, and in Section 3 we will give some lemmas and then prove the main results.

Main Results
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 Λ {t : λ t > 0} {t : P τ 1 ≤ t > 0}.

Related Results and Motivations
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, Aleškevičien ė et al. 2 considered the standard compound renewal risk model with condition 1.7 and showed that Inspired by the above results, we will further discuss some issues as follows: 1 to cancel the moment condition on {θ i , i ≥ 1}, namely, Eθ In the paper, we will answer the four issues directly, and then we obtain our main results in the next section.

Main Results
For the main results of this paper, we now state some conditions which are that of Wang et al. 9 .
Condition 2. The interarrival times {θ i , i ≥ 1} are WOD r.v.s and there exists a positive and nondecreasing function g x such that g x ↑ ∞, x −k g x ↓ for some 0 < k < 1, Eθ 1 g θ 1 < ∞, and max{g U n , g L n } ≤ g n for all n ≥ 1, where x −k g x ↓ means that x −k 1 g x 1 ≥ Cx −k 2 g x 2 for all 0 ≤ x 1 < x 2 < ∞ and some finite constant C > 0.
The first main result of this paper is the following.
So, Conditions 1-4 in Theorems 2.1 and 2.2 can be replaced by 2.7 .
Advances in Decision Sciences 9

Proofs of Main Results
In this section we will give the proofs of our main results, for which we need some following lemmas.
Lemma 3.1.If {ξ i , 1 ≤ i ≤ n} are n WUOD and nonnegative r.v.s with distributions V i ∈ L∩D, 1 ≤ i ≤ n, respectively, then for any fixed 0 < a ≤ b < ∞,  then for any γ > 0, there exists a constants C C γ such that holds for all x ≥ γn and all n ≥ 1.
Proof.By Proposition 1.2 and along the same lines of the proof of Theorem 3.1 of Tang 26 with slight modifications, we can derive that, for some positive integer m,

Advances in Decision Sciences
From Lemma 3.2, we have sup 1≤n≤m,x≥γn Combining 3.5 and 3.6 , there exists a constant C γ > 0 such that 3.4 holds for all x ≥ γn and all n ≥ 1.
The following lemma discusses the strong law of large numbers for widely dependent r.v.s, which is due to Wang and Cheng 27 .Lemma 3.5.Let {ξ i , i ≥ 1} be a sequence of identically distributed and real-valued r.v.s with distribution V and finite mean a and satisfy one of the Conditions 1-4 with {θ i , i ≥ 1} replaced by {ξ i , i ≥ 1}, where for Conditions 2 and 3 one further assumes that 3.3 holds.Let η be nonnegative integer-valued r.v. with distribution U and finite mean b, independent of {ξ i , i ≥ 1}.Assume that V x ∼ cU x for some c ∈ 0, ∞ .
and the conditions of Lemma 3.4 are valid, then 3 Let no assumption be made on the interrelationship between V and U.If V ∈ L∩D, U ∈ C, and the conditions of Lemma 3.4 are still valid, then relation 3.8 still holds.
Proof.Because η has finite mean, there exists a large integer m 0 > 0 such that, for any fixed ε > 0, it holds that 1 First consider the case that 0 < c < ∞.Clearly, U ∈ C implies V ∈ C. For any x > 0 and any δ ∈ 0, 1 , we have

3.10
For K 1 , by Lemma 3.2 it follows that For K 2 , since n < 1 − δ x/a and V ∈ C ⊂ D, we obtain by Lemma 3.3 that

3.12
where γ δa/ 1 − δ , C γ and C γ are two constants only depending on γ.Hence, applying 3.9 , Lemma 3.2, and the dominated convergence theorem can yield that

3.13
For K 3 , since δ ∈ 0, 1 can be arbitrarily close to 0, we see by U ∈ C that 3.14 Substituting 3.11 -3.14 into 3.10 and considering the arbitrariness of ε > 0, we derive that

3.15
On the other hand, we note that

3.16
For K 1 , by Lemma 3.2 and 3.9 , we get For K 4 , by Lemma 3.4 we find that which, along with V ∈ C and the arbitrariness of δ ∈ 0, 1 , leads to
Next we turn to the case that c 0, namely, V x o U x .According to Lemma 4.4 of Fa ÿ et al. 29 , there exists a nondecreasing slowly varying function L x → ∞ such that V x o U x /L x , which results in that for some x 0 > 0, V x ≤ U x /L x ≤ 1, for all x ≥ x 0 .

3.21
Advances in Decision Sciences 13 Define

3.22
where It is easy to verify that {ξ * i , i ≥ 1} are still WUOD and identically distributed r.v.s with common distribution V * ∈ C. By the definition of V * , we know that ξ i ≤ ξ * i , i ≥ 1, and then a ≤ Eξ * i a * < ∞.Thus, K 1 K 2 in 3.10 is divided into three parts as

3.24
Clearly, {ξ * i , i ≥ 1} are such that the conditions of Lemmas 3.2 and 3.3 hold, then 3.11 and 3.13 can still hold with K 1 , K 2 , and {ξ i , i ≥ 1} replaced by K 1 , K 2 , and {ξ * i , i ≥ 1}.So, we deduce by For K 2 , it follows from Lemma 3.4 that Then, by U ∈ C ⊂ D we have

3.28
Again by 3.16 and 3.19 , it is seen that
2 Now we deal with the case that c ∞, namely, U x o V x .Apparently, when V ∈ L ∩ D, we can derive by Lemmas 3.2 and 3.3 that 3.11 and 3.13 still hold.As for K 3 , by U x o V x and V ∈ L ∩ D, we know that

3.32
For V ∈ L ∩ D, by Lemma 3.2 we also get 3.17 .As for K 4 , arguing as 3.19 and 3.31 , we still have that

3.34
Similarly to the derivation of 3.30 , by U x o V x and V ∈ L ∩ D we still see that U x/a o bV x .This, along with 3.32 and 3.34 , gives relation 3.8 immediately.
3 According to the proof of 2 , we know that if V ∈ L ∩ D, then 3.11 and 3.13 hold.While from the proof of 1 , we have 3.14 if U ∈ C. Hence, under the conditions of 3 , we obtain 3.15 .
On the other hand, from the proof of 2 , we can get 3.17 when V ∈ L ∩ D. Again by the proof of 1 , relation 3.19 also holds for U ∈ C. So, 3.20 is proved under the conditions of 3 .As a result, we show 3.8 directly.
The next two lemmas will give some results of the renewal risk model, which is the compound renewal risk model with N 1 N 2 • • • 1.
Condition 5.For α in 2.5 , there exist t 0 ∈ Λ ∩ 0, ∞ and f in 1.12 such that f x α P θ Now we prove the main results as follows.
Proof of Theorem 2.1.Clearly, if F ∈ L ∩ D then F satisfies Condition 5.In fact, the assumption F ∈ L ∩ D indicates that the tail of F behaves essentially like a power function, thus there exists q > 0 such that x q F x → ∞.Take f x x p for any 0 < p < 1, which satisfies 1.12 .So, for any s > 0, e −sx p /F x → 0 and Condition 5 holds.
First, consider Theorem 2.1 1 .For the case that 0 ≤ c < ∞, we can obtain relation 2.4 by 1.5 and Lemmas 3.5 1 and 3.6, and going along the similar ways to that of For two positive functions a • and b • , we write a x b x if lim sup a x /b x ≤ 1, write a x b x if lim inf a x /b x ≥ 1, write a x ∼ b x if both, write a x o b x if lim a x /b x 0. For two positive bivariate functions a •, • and b •, • , we say that relation a x, t ∼ b x, t holds uniformly for all t

p 1 < 1 <
∞ for some p > J G 1, and Eθ p ∞ for some p > J F 1; 2 to extend partially the class C or R to the class L ∩ D;3 to discuss the case when {X n i , i ≥ 1} are WUOD and {θ i , i ≥ 1} are WLOD; 4 to drop the interrelationships between F and G and investigate the case when both F and G are heavy tailed.

Lemma 3 . 4 .Proof.
Let {ξ i , i ≥ 1} be a sequence of real-valued r.v.s with finite mean a > 0 and satisfy one of the Conditions 1-4 with {θ i , i ≥ 1} replaced by {ξ i , i ≥ 1}.Then lim Follow Theorem 1 of Matula 28 , Theorem 1.4, and the proofs of Theorems 1.1 and 1.2 of Wang and Cheng 27 , respectively.The lemma below gives the tail behavior of random sum, which extends the results of Aleškevičien ė et al. 2 and Zhang et al. 3 .
claim sizes and their number caused by nth accident are {X ≥ 1, n ≥ 1} are nonnegative and identically distributed r.v.s with common distribution F and finite mean μ, and {X 1} are mutually independent for all n / m, n, m ≥ 1, while {N n , n ≥ 1} are independent, identically distributed i.i.d., and positive integer-valued r.v.s with common distribution G and finite mean ν.The sequences {θ i , i ≥ 1}, {X n i , i ≥ 1, n ≥ 1}, and {N n , n ≥ 1} are mutually independent.Denote the arrival times of the nth accident by τ n n i 1 θ i , n ≥ 1, which can form a nonstandard renewal counting processN t sup{n ≥ 1, τ n ≤ t}, t ≥ 0, 1.1with mean function λ t EN t .Hence the total claim amount at time τ n and the total claim amount up to time t ≥ 0 are, respectively, and {N n , n ≥ 1} are i.i.d.r.v.s, then the model is reduced to the standard compound renewal risk model introduced by Tang et al. 1 , if N 1 N 2 • • • 1, then the model is the renewal risk model, see Tang 6 , Leipus and Šiaulys 7 , Yang et al. 8 , and Wang et al. 9 , among others.
H * 1The interarrival times {θ i , i ≥ 1} are nonnegative, identically distributed and WLOD r.v.s with finite mean λ −1 .1} are nonnegative, identically distributed and WUOD r.v.s, and the other statements of Assumption H 2 are still valid.
∈ R −α , G x ∼ cF x for some constant c > 0, and Eθ p 1 < ∞ for some p > α 1, then it holds uniformly for all t ∈ Λ that 1} caused by nth accident are negatively associated and obtained a unified form of ψ x, t as follows: let F x ∼ cG x for c ∈ 0, ∞ , and one of the conditions below holds: i for c 0, G ∈ C, and Eθ p 1 < ∞ for some p > J G 1; ii for c > 0, F ∈ C, and Eθ p 1 < ∞ for some p > J F 1; then it holds uniformly for all t ∈ Λ that Observe the results of Zhang et al. 3 especially extended case iii of Aleškevičien ė et al. 2 .Also, Lin and Shen 4 considered a generalized compound renewal risk model with {X 1} satisfying one type of asymptotically quadrant subindependent structure and also obtained the same relations 2.2 , 2.3 , and 2.4 as that of Aleškevičien ė et al. 2 .
n i , i ≥ 19te that there do exist some WLOD r.v.s satisfying condition 2.5 , see Wang et al.19.In the second main result below, we discuss the random time ruin probability, which requires another assumption.Let τ be nonnegative r.v. and independent of the sequences {θ i , i ≥ 1}, {X n 1 If F x ∼ cG x for some c ∈ 0, ∞ , additionally, for c 0 and G ∈ C; for 0 < c < ∞ and F ∈ C; for c ∞ and F ∈ L ∩ D, then relation 2.4 holds uniformly for all t ∈ Λ. 2 If F ∈ L ∩ D and G ∈ C, relation 2.4 holds uniformly for all t ∈ Λ. i , i ≥ 1, n ≥ 1}, and {N n , n ≥ 1}.Define Δ {τ : P τ ∈ Λ > 0}.Theorem 2.2.Under conditions of Theorem 2.1 and Assumption H 4 , one has 1 if F x ∼ cG x for some c ∈ 0, ∞ , additionally, for c 0 and G ∈ C; for 0 < c < ∞ and F ∈ C; for c ∞ and F ∈ L ∩ D, then it holds uniformly for all t ∈ Δ that 2 if F ∈ L ∩ D and G ∈ C, then relation 2.6 still holds uniformly for all t ∈ Δ.
1 holds uniformly for all c n c 1 , c 2 , . . ., c n ∈ a, b n .Proof.See Lemma 3.1 of Gao et al. 25 .Particularly, let c 1 c 2 • • • c n 1 in Lemma 3.2, we have a lemma below.