# Uniform Asymptotics for the Finite-Time Ruin Probability of a Time-Dependent Risk Model with Pairwise Quasiasymptotically Independent Claims

Authors: Qingwu Gao
Journal: ISRN Probability and Statistics
 (2012)

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## Abstract

We consider a generalized time-dependent risk model with constant interest force, where the claim sizes are of pairwise quasiasymptotical independence structure, and the claim size and its interclaim time satisfy a dependence structure defined by a conditional tail probability of the claim size given the interclaim time before the claim occurs. As the claim-size distribution belongs to the dominated variation class, we establish some weakly asymptotic formulae for the tail probability of discounted aggregate claims and the finite-time ruin probability, which hold uniformly for all times in a relevant infinite interval.


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## Body

## 1. Introduction

In the paper we will consider a generalized risk model of an insurance company, in which the claim sizes{Xi,i≥1} are nonnegative, identically distributed, but not necessarily independent random variables (r.v.s) with common distribution F and generic r.v. X, and their interarrival times {θi,i≥1} are other independent, identically distributed (i.i.d.), and nonnegative r.v.s with generic r.v. θ. To avoid triviality, X and θ are assumed not to be degenerate at 0. Denote the claim arrival times by τ0=0, τn=∑i=1n‍θi, n≥1, which constitute a renewal counting process as follows:
(1.1)N(t)=sup{n≥1,τn≤t},t≥0,
with a finite-mean function λ(t)=EN(t)=∑i=1∞‍P(τi≤t), t≥0. Assume that, for every i≥2, Xi and τi-1 are mutually independent. Meanwhile, as mentioned by Wang [1], the total amount of premiums accumulated before t≥0, denoted by C(t) with C(0)=0 and C(t)<∞ almost surely for any fixed t>0, is a nonnegative and nondecreasing stochastic process. Let r≥0 be the constant interest force and x≥0 be the insurer's initial reserve. Hence, the total reserve up to t≥0 of the insurance company, denoted by Ur(t), satisfies as following:
(1.2)Ur(t)=xert+∫0t‍er(t-s)C(ds)-∫0t‍er(t-s)S(ds),
and the discounted aggregate claims up to t≥0 are expressed as
(1.3)Dr(t)=∫0t‍e-rsS(ds)=∑i=1∞Xie-rτi1{τi≤t},
where S(t)=∑i=1N(t)‍Xi is the aggregate claim amount before t≥0 with S(t)=0 if N(t)=0. As usual, the ruin probability within a finite time t>0 is defined by
(1.4)ψr(x,t)=P(Ur(s)<0for some0≤s≤t),
and the infinite-time ruin probability is
(1.5)ψr(x,∞)=P(Ur(t)<0for some0≤t<∞).For the renewal risk model with i.i.d. claim sizes{Xi,i≥1} and i.i.d. interarrival times {θi,i≥1}, in which {Xi,i≥1} and {θi,i≥1} are mutually independent, there are many related works on ruin theory with a constant interest force r>0, for example, see Klüppelberg and Stadtimüller [2], Kalashnikov and Konstantinides [3], Konstantinides et al. [4], Tang [5, 6], and  Hao and Tang [7], among others. However, the independence assumptions above are made mainly not for practical relevance but for theoretical interest.In recent years, various extensions to the renewal risk model have been proposed to appropriately relax these independence assumptions. Generally, there are two directions to discuss the extensions. One is that a certain dependence structure is imposed on the claim sizes{Xi,i≥1} and/or their interarrival times {θi,i≥1}, but {Xi,i≥1} are assumed to be independent of {θi,i≥1}. See, for example, Chen and Ng [8], Li et al. [9], Yang and Wang [10], Wang et al. [11], and Liu et al. [12], and references therein. The other is that the claim size X and its interarrival time θ follow a certain dependence structure, but {(Xi,θi),i≥1} are i.i.d. random pairs. In this direction, many researchers considered some ruin-related problems of a generalized risk model with a certain dependence between X and θ when the claim-sized distribution is light tailed, for example, Albrecher and Teugels [13], Boudreault et al. [14], Cossette et al. [15], and Badescu et al. [16]. Besides, Asimit and Badescu [17] introduced a general dependence structure for (X,θ) and presented the tail behavior of discounted aggregate claims Dr(t) for the compound Poisson model with constant interest force and heavy-tailed claims. Under the dependence structure of Asimit and Badescu [17], Li et al. [18] extended the tail behavior of Dr(t) to the renewal risk model.The dependence structure betweenX and θ introduced by Asimit and Badescu [17] satisfies that the relation
(1.6)P(X>x∣θ=t)~F¯(x)h(t),t≥0,
holds for some measurable function h(·):[0,∞)↦(0,∞), where the symbol ~ means that the quotient of both sides tends to 1 as x→∞. When t is not a possible value of θ, the conditional probability in (1.6) is understood as an unconditional one, and then h(t)=1. If relation (1.6) holds uniformly for all t∈[0,∞), then, by conditioning on τi-1 and θi, i≥1, it holds uniformly for all t∈[0,∞) that
(1.7)P(Xie-rτi1{τi≤t}>x)~∫0t‍∫0t-v‍P(Xie-r(u+v)>x)P(τi-1∈du)h(v)G(dv)=P(Xie-r(τi-1+θ*)1{τi-1+θ*≤t}>x),
where θ* is a r.v. independent of X and θ, with a proper distribution given by
(1.8)P(θ*∈dv)=h(v)G(dv).Remark 1.1.
Note that the general dependence structure defined by (1.6) can cover both positive and negative dependence and is also easily verifiable for many common bivariate copulas, which can be found in Li et al. [18]. Furthermore, practitioners in insurance industry often meet the following situations: for autoinsurance or fire insurance, if the claim interarrival time is longer, then more measures can be taken to reduce the forthcoming losses of property, while if the deductible of the insured is raised, then the claim interarrival time will increase since some small claims can avoid. These situations tell us that the claim size and its interarrival time are interacted on each other, and the dependence structure defined by (1.6) is realistic in actuarial environments.Based on the two study directions above to extend the renewal risk model, this paper will consider a more generalized risk model with the claim sizes following some dependence structure as well as the random pair(X,θ) satisfying relation (1.6) and establish the weakly asymptotic formulae for the tail probability of discounted aggregate claims and the finite-time ruin probability, which hold uniformly for all times in a relevant infinite interval. The method used in the paper is different from that in the above mentioned literatures, and the obtained results can extend and improve some existing results.The rest of this paper is organized as follows: Section2 will present the main results of this paper after preparing some preliminaries, and Section 3 will give some lemmas which are helpful to prove our main results in Section 4.


## 2. Preliminaries and Main Results

All limit relationships in the paper are forx→∞ unless mentioned otherwise. For two positive functions a(·) and b(·), we write a(x)=O(1)b(x) if limsupa(x)/b(x)=c<∞, write a(x)≲b(x) or b(x)≳a(x) if c≤1, write a(x)~b(x) if a(x)≲b(x) and b(x)≲a(x), write a(x)=o(1)b(x) if c=0, and write a(x)≍b(x) if a(x)=O(1)b(x) and b(x)=O(1)a(x). Further, for two positive bivariate functions a(·,·) and b(·,·), we write a(x,t)~b(x,t) uniformly for all t∈Δ≠∅ if limsupt∈Δ|a(x,t)/b(x,t)-1|=0.As everyone knows, in the insurance industry how to model the dangerous claims is one of the main worries of the practicing actuaries, and actually most practitioners choose the claim-size distribution from the heavy-tailed distribution class, one of which is the dominated variation class. By definition, a distributionF belongs to the dominated variation class, denoted by F∈𝒟, if F¯(xy)=O(1)F¯(x) for all y>0, where F¯(x)=1-F(x); belongs to the extended regular variation (ERV) class if there exist some 0<α≤β<∞ such that y-β≤F¯*(y)≤F¯*(y)≤y-α for all y≥1, where F¯*(y)=liminfF¯(xy)/F¯(x) and F¯*(y)=limsupF¯(xy)/F¯(x); belongs to the consistent variation class, denoted by F∈𝒞, if LF=limy↘1F¯*(y)=1; belongs to the subexponential class, denoted by F∈𝒮, if F¯(x)>0 for all x>0 and F*2¯(x)~2F¯(x), where F*2 denotes the 2-fold convolution of F. Note that if F∈𝒮 then F is long tailed, denoted by F∈ℒ and characterized by F¯(x+y)~F¯(x) for all y≠0.For a distributionF and any y>0, we set JF+=-limy→∞logF¯*(y)/logy and JF-=-limy→∞logF¯*(y)/logy. It is well known that the following relationships hold, namely:
(2.1)ERV⊂𝒞⊂ℒ∩𝒟⊂𝒮⊂ℒ,𝒟⊄ℒ,ℒ⊄𝒟.
For more details of heavy-tailed distributions and their applications to insurance and finance, the readers are referred to Bingham et al. [19] and Embrechts et al. [20]. Additionally, some reviews on the class 𝒟 and its application can be found in Shneer [21], Wang and Yang [22], and others.Following the first trend to extend the renewal risk model, in the paper we will discuss a risk model with the claim sizes satisfying the following dependence structure.Definition 2.1.
Say that the r.v.s{ξi,i≥1} with their respective distributions Vi,i≥1 are pairwise quasiasymptotically independent if
(2.2)P(ξi>x,ξj>x)=o(1)(Vi¯(x)+Vj¯(x))fori≠j,i,j≥1.Remark 2.2.
The term “pairwise quasiasymptotic independence” is borrowed from Chen and Yuen [23]. Clearly, if {ξi,i≥1} are identically distributed, relation (2.2) is equivalent to
(2.3)P(ξi>x,ξj>x)=o(1)P(ξi>x)fori≠j,i,j≥1,
which means that {ξi,i≥1} are pairwise asymptotically independent or bivariate upper tail independent; see Zhang et al. [24] and Gao and Wang [25]. Also remark that the sequence of pairwise quasiasymptotically independent r.v.s can cover the sequence of widely upper orthant dependent/widely lower orthant dependent r.v.s (see Wang et al. [11]), extended negatively dependent r.v.s (see Liu [26]), negatively upper orthant dependent/negatively lower orthant dependent r.v.s (NUOD/NLOD, see Block et al. [27]), pairwise negatively quadrant dependent r.v.s (NQD, see Lehmann [28]), and even it can cover some sequences of positive dependent r.v.s.For statement convenience, we denote byΛ the set of all t such that 0<λ(t)≤∞. Let t_=inf{t:P(τ1≤t)>0}, it is clear that
(2.4)Λ={[t_,∞],ifP(τ1=t_)>0;(t_,∞],ifP(τ1=t_)=0.
Also we write ΛT=[0,T]∩Λ for any finite T∈Λ.Under the case that the claim sizes and/or interarrival times follow some dependence structure, Li et al. [9] obtained a weakly asymptotic formula for the finite-time ruin probability as follows.Theorem 2.3.
Consider the insurance risk model introduced in Section1, in which the claim sizes are pairwise NQD with common distribution F∈𝒟 such that JF->0, the interarrival times {θi,i≥1} are NLOD, and the premium process {C(t),t≥0} is a deterministic linear function. If {Xi,i≥1} and {θi,i≥1} are mutually independent, then, for every fixed t∈Λ,
(2.5)LF∫0t‍F¯(xers)dλ(s)≲ψr(x,t)≲LF-2∫0t‍F¯(xers)dλ(s).In addition, if{C(t),t≥0} is a general stochastic process and {N(t),t≥0} is a delayed renewal counting process, Yang and Wang [10] also gave the formula (2.5) for ψr(x,t).Following the second extension direction, Li et al. [18] have investigated the tail behavior of discounted aggregate claims Dr(t) for the renewal risk model with X and θ meeting the dependence structure defined by (1.6).Theorem 2.4.
Consider the discounted aggregate claims defined in (1.3) with r>0, in which {(Xi,θi),i≥1} are i.i.d. random pairs, and relation (1.6) holds uniformly for all t∈Λ. If F∈ERV(-α,-β) for some 0<α≤β<∞, and inf0≤t≤t*h(t)>0 for some t*∈Λ, then
(2.6)P(Dr(t)>x)~∫0t‍F¯(xers)dλ~(s)
holds uniformly for t∈Λ, where
(2.7)λ~(t)=∑i=1∞‍P(τi-1+θ*≤t).Inspired by the results of Theorems2.3 and 2.4, in this paper we will further discuss the following issues.(1)
For the structure assumptions on the claim sizes, we will study the more general case of pairwise quasiasymptotically independence assumption.(2)
We will extend the mutual independence between the claim sizes and their inter-times to a general dependence structure defined by relation (1.6).(3)
We will extend the conditionF∈ERV of Theorem 2.4 to the conditions F∈𝒟 (or F∈𝒞).(4)
We will cancel the condition thatinf0≤t≤t*h(t)>0 for some t*∈Λ of Theorem 2.4.(5)
We do not confine this paper only to the case thatr>0, but discuss the case that r≥0.(6)
We will discuss the case when the premium process{C(t),t≥0} is not necessarily independent of {Xi,i≥1} and {N(t),t≥0}.The following are the main results of this paper, among which the first two theorems discuss the tail behavior of the discounted aggregate claimsDr(t) described by (1.3).Theorem 2.5.
Consider the discounted aggregate claims (1.3) described in Section 1 with r≥0. If the claim sizes {Xi,i≥1} are pairwise quasiasymptotically independent r.v.s with common distribution F∈𝒟 such that JF->0, and relation (1.6) holds uniformly for all t∈ΛT, then it holds uniformly for all t∈ΛT that
(2.8)∫0t‍F¯(xers)dλ~(s)≲P(Dr(t)>x)≲LF-2∫0t‍F¯(xers)dλ~(s).Additionally ifF∈𝒞, then relation (2.6) holds uniformly for all t∈ΛT.The second theorem extends the set over which relations (2.6) and (2.8) hold uniformly to the whole set Λ. It is well known that if r=0, then Dr(t)→∞ almost surely as t→∞, and hence it is not impossible to establish the uniformity of (2.6) and (2.8) for all t∈Λ. So we assume that r>0 in the following result.Theorem 2.6.
Consider the discounted aggregate claims (1.3) described in Section 1 with r>0. If the claim sizes {Xi,i≥1} are pairwise quasiasymptotically independent r.v.s with common distribution F∈𝒟 such that JF->0, and relation (1.6) holds uniformly for all t∈Λ, then relation (2.8) still holds uniformly for all t∈Λ. Further assume that F∈𝒞, then relation (2.6) still holds uniformly for all t∈Λ.In what follows, we will deal with the asymptotic behavior of the finite-time and infinite-time ruin probabilities, where we will discuss two cases: one is that the premium process{C(t),t≥0} is independent of {Xi,i≥1} and {N(t),t≥0}, and the other is that {C(t),t≥0} is not necessarily independent of {Xi,i≥1} or {N(t),t≥0}. For later use, we write the discounted value of premiums accumulated before time t as
(2.9)C~(t)=∫0t‍e-rsC(ds).
Clearly, by the conditions on C(t), it holds that 0≤C~(t)<∞ almost surely for any fixed 0<t<∞.Theorem 2.7.
Consider the insurance risk model introduced in Section1 with r≥0. Under the conditions of Theorem 2.5, it holds uniformly for t∈ΛT that
(2.10)LF∫0t‍F¯(xers)dλ~(s)≲ψr(x,t)≲LF-2∫0t‍F¯(xers)dλ~(s),
if either of the following conditions holds:(1)
the premium process{C(t),t≥0} is independent of {Xi,i≥1} and {N(t),t≥0};(2)
the discounted value of premiums accumulated by timet satisfies that
(2.11)P(C~(t)>x)=o(1)F¯(x).Particularly, if F∈𝒞, it holds uniformly for t∈ΛT that
(2.12)ψr(x,t)~∫0t‍F¯(xers)dλ~(s).Applying Theorem2.7, we now propose a corollary for the case when r=0.Corollary 2.8.
Consider the insurance risk model in Section1 with r=0; if the conditions of Theorem 2.7 are valid, then it holds uniformly for all t∈ΛT that
(2.13)LFF¯(x)λ~(t)≲ψ0(x,t)≲LF-2F¯(x)λ~(t),(2.14)κ-1LF∫xx+κλ~(t)‍F¯(y)dy≲ψ0(x,t)≲κ-1LF-3∫xx+κλ~(t)‍F¯(y)dy,
where κ is any positive number. Further assume that F∈𝒞, and then
(2.15)ψ0(x,t)~F¯(x)λ~(t)~κ-1∫xx+κλ~(t)‍F¯(y)dy
holds uniformly for all t∈ΛT.Theorem 2.9.
Consider the insurance risk model introduced withr>0. Under the conditions of Theorem 2.6, relation (2.10) holds uniformly for t∈Λ, if either of the following conditions holds:(1)
the premium process{C(t),t≥0} is independent of {Xi,i≥1} and {N(t),t≥0};(2)
the total discounted amount of premiums satisfies the following:(2.16)0≤C~(∞)<∞almostsurely,P(C~(∞)>x)=o(1)F¯(x).Particularly, if F∈𝒞, then relation (2.12) still holds uniformly for t∈Λ.According to the uniformity ofψr(x,t) for all t∈Λ in Theorem 2.9, we can immediately derive the corresponding result on the infinite-time ruin probability ψr(x,∞).Corollary 2.10.
Under conditions of Theorem2.9, one has
(2.17)LF∫0∞‍F¯(xers)dλ~(s)≲ψr(x,∞)≲LF-2∫0∞‍F¯(xers)dλ~(s).
If F∈𝒞, then
(2.18)ψr(x,∞)~∫0∞‍F¯(xers)dλ~(s).Remark 2.11.
We would like to make some explanations of conditionsF∈𝒟 with JF->0 in the main results. Wang and Yang [22] proposed the following assertions.(i)
𝒟=𝒟1∪𝒟2∪𝒟3, where
(2.19)𝒟1={Fon[0,∞):F¯*(y)=1∀y>1},𝒟2={Fon[0,∞):0<F¯*(y)<F¯*(y)=1∀y>1},𝒟3={Fon[0,∞):0<F¯*(y)≤F¯*(y)<1∀y>1},
and 𝒟1, 𝒟2, and 𝒟3 are pairwise disjoint sets.(ii)
F∈DwithJF->0⇔F∈𝒟3.(iii)
Denote that(2.20)𝒟(-α,-β,A,B)={Fon[0,∞):By-β≤F¯*(y)≤F¯*(y)≤Ay-α∀y>1},
where 0≤α≤β<∞, 0<B≤1≤A<∞, and then
(2.21)𝒟=⋃0≤α≤β<∞,0<B≤1≤A<∞‍D(-α,-β,A,B),𝒟3=⋃0<α≤β<∞,0<B≤1≤A<∞‍D(-α,-β,A,B).In the particular case where A=B=1, the class ERV⊂D3 and this inclusion are proper. For example, the Peter and Paul distribution (see Example 1.4.2 in Embrechts et al. [20]) belongs to D3, but it does not belong to L; thus, it does not belong to the class ERV.Remark 2.12.
By the expression of (2.7), one can easily see that λ~(t) is exactly the mean function of a delayed renewal process constituted by {θ*,θi,i≥2}.Remark 2.13.
The Lemma3.3 below tells us that Xie-rτi1{τi≤t} and Xje-rτj1{τj≤t} are quasiasymptotically independent for all t∈ΛT and every fixed i≠j≥1 under the following assumptions: the claim sizes {Xi,i≥1} are pairwise quasiasymptotically independent with common distribution F∈𝒟, and their interarrival times {θi,i≥1} are independent, identically distributed and such that (1.6) holds uniformly for all t∈ΛT. Hence, the dependence structures among claim sizes and that between claim size and its interarrival time in the paper are technically feasible.Remark 2.14.
In Theorems2.7–2.9 and Corollaries 2.8–2.10, the independence between the premium process and the claim process in condition 1 has been extensively considered by Wang [1], Yang and Wang [10], Wang et al. [11], Liu et al. [12], and many others, while condition 2, which does not require the independence between the premium process and the claim process, allows for a more realistic case that the premium rate varies as a deterministic or stochastic function of the insurer's current reserve, as that considered by Petersen [29], Michaud [30], Jasiulewicz [31], and Tang [5].


## 3. Some Lemmas

In order to prove the main results, we need the following lemmas, among which the first lemma is a combination of Proposition 2.2.1 of Bingham et al. [19] and Lemma 3.5 of Tang and Tsitsiashvili [32].Lemma 3.1.
For a distributionF on (-∞,∞), the following assertions hold:(1)
F∈𝒟⇔LF>0⇔JF+<∞,(2)
IfF∈𝒟, then for any p1<JF- and any p2>JF+, there are positive numbers Ci and Di, i=1,2, such that
(3.1)F¯(y)F¯(x)≥C1(xy)p1∀x≥y≥D1,(3.2)F¯(y)F¯(x)≤C2(xy)p2∀x≥y≥D2.Lemma 3.2.
Consider the insurance risk model introduced in Section1 with r≥0 and F∈𝒟, where a generic pair (X,θ) is such that relation (1.6) holds uniformly for all t∈ΛT, and then(1)
the distribution of theXie-rτi1{τi≤t} belongs to the class 𝒟 for every fixed i≥1 and all t∈ΛT; moreover, if F∈𝒞, then the distribution of the Xie-rτi1{τi≤t} still belongs to the class 𝒞 for every fixed i≥1 and all t∈ΛT;(2)
for every fixedi≥1 and all t∈ΛT, it holds that
(3.3)F¯(x)≍P(Xie-rτi1{τi≤t}>x).Proof.
(1) IfF∈𝒟, then we have from (1.7) and Theorem 3.3(ii) of Cline and Samorodnitsky [33] that, for all y>0,
(3.4)limsupx→∞P(Xie-rτi1{τi≤t}>xy)P(Xie-rτi1{τi≤t}>x)=limsupx→∞P(Xie-r(τk-1+θ*)1{τk-1+θ*≤t}>xy)P(Xie-r(τk-1+θ*)1{τk-1+θ*≤t}>x)<∞
holds for every fixed i≥1 and all t∈ΛT, which implies that the distribution of the Xie-rτi1{τi≤t} belongs to the class 𝒟 for every fixed i≥1 and all t∈ΛT. Moreover, if F∈𝒞, then by (1.7) and Lemma 2.5 of Wang et al. [34], we derive that, for every fixed i≥1 and all t∈ΛT,
(3.5)limy↘1liminfx→∞P(Xie-rτi1{τi≤t}>xy)P(Xie-rτi1{τi≤t}>x)=limy↘1liminfx→∞P(Xie-r(τk-1+θ*)1{τk-1+θ*≤t}>xy)P(Xie-r(τk-1+θ*)1{τk-1+θ*≤t}>x)=1,
which also implies that the distribution of the Xie-rτi1{τi≤t} belongs to the class 𝒟 for every fixed i≥1 and all t∈ΛT.
(2) ByF∈𝒟, (1.7) and Theorem 3.3(iv) of Cline and Samorodnitsky [33], we can prove that (3.3) holds.Lemma 3.3.
Under the conditions of Theorem2.5, if F∈𝒟, then Xie-rτi1{τi≤t} and Xje-rτj1{τj≤t} are still quasiasymptotically independent for all t∈ΛT and every fixed i≠j≥1.Proof.
Without loss of generality, we assume thatj>i≥1. By conditioning on τi-1 and θi, i≥1, and using (1.6) and (1.7), it holds for all t∈ΛT that
(3.6)P(Xie-rτi1{τi≤t}>x,Xje-rτj1{τj≤t}>x)=∫0t∫0t-vP(Xie-r(u+v)>x,Xje-r(u+v+θi+1+⋯+θj)1{θi+1+⋯+θj≤t-(u+v)}>x∣τi-1=u,θi=v)×P(τi-1∈du)G(dv)=∫0t∫0t-vP(Xie-r(u+v)>x∣θi=v)×P(Xje-r(u+v+θi+1+⋯+θj)1{θi+1+⋯+θj≤t-(u+v)}>x∣Xie-r(u+v)>x)P(τi-1∈du)G(dv)~∫0t∫0t-vP(Xie-r(u+v)>x,Xje-r(u+v+θi+1+⋯+θj)1{θi+1+⋯+θj≤t-(u+v)}>x)×P(τi-1∈du)P(θ*∈dv)=P(Xie-r(τi-1+θ*)1{τi-1+θ*≤t}>x,Xje-r(τj-1+θ*)1{τi-1+θ*≤t}>x).
Denote Wi=e-r(τi-1+θ*)1{τi-1+θ*≤t} and Wj=e-r(τj-1+θ*)1{τj-1+θ*≤t}, which are clearly independent of (Xi,Xj). Thus, by (2.2) and (1.7), we will complete this proof if we can show that
(3.7)P(WiXi>x,WjXj>x)=o(1)(P(WiXi>x)+P(WjXj>x))fori≠j≥1.
Let Hi and Hj be the distributions of Wi and Wj, respectively, and H(s,t) be the joint distribution of Wi and Wj, then we see that
(3.8)P(WiXi>x,WjXj>x)≤∫‍∫{0<s≤t≤1}‍P(Xi>xt,Xj>xt)dG(s,t)+∫‍∫0<t<s≤1‍P(Xi>xs,Xj>xs)dG(s,t)=A1(x)+A2(x).
For A1(x), by (2.2), it follows that
(3.9)A1(x)≤∫01‍P(Xi>xt,Xj>xt)dH2(t)≤(P(Xi>x)+P(Xj>x))∫01‍P(Xi>(x/t),Xj>(x/t))P(Xi>(x/t))+P(Xj>(x/t))dH2(t)=o(1)(P(Xi>x)+P(Xj>x)).
Then, by Theorem 3.3(iv) of Cline and Samorodnitsky [33], we have
(3.10)A1(x)=o(1)(P(Xi>x)+P(Xj>x))=o(1)(P(WiXi>x)+P(WjXj>x))fori≠j≥1.
Similarly, we also have
(3.11)A2(x)=o(1)(P(WiXi>x)+P(WjXj>x))fori≠j≥1.
Therefore, we attain (3.7), and then we complete the proof.Lemma 3.4.
If the claim sizes{Xi,i≥1} are pairwise quasiasymptotically independent r.v.s with common distribution F∈𝒟, and relation (1.6) holds uniformly for all t∈ΛT, and then, for r≥0 and for every fixed n≥1, it holds uniformly for all t∈ΛT that
(3.12)∑i=1n‍P(Xie-rτi1{τi≤t}>x)≲P(∑i=1n‍Xie-rτi1{τi≤t}>x)≲LF-1∑i=1n‍P(Xie-rτi1{τi≤t}>x).
Additionally, if F∈𝒞, then it holds uniformly for all t∈ΛT that
(3.13)P(∑i=1n‍Xie-rτi1{τi≤t}>x)~∑i=1n‍P(Xie-rτi1{τi≤t}>x).Proof.
Clearly, for every fixedn≥1 and all t∈ΛT, we have
(3.14)P(∑i=1n‍Xie-rτi1{τi≤t}>x)≥∑i=1n‍P(Xie-rτi1{τi≤t}>x)-∑1≤i≠j≤n‍P(Xie-rτi1{τi≤t}>x,Xje-rτj1{τj≤t}>x)=∑i=1n‍P(Xie-rτi1{τi≤t}>x)-A3.
Form Lemma 3.3, we see that
(3.15)limsupx→∞A3∑i=1nP(Xie-rτi1{τi≤t}>x)≤limsupx→∞∑1≤i≠j≤n‍P(Xie-rτi1{τi≤t}>x,Xje-rτj1{τj≤t}>x)P(Xie-rτi1{τi≤t}>x)+P(Xje-rτj1{τj≤t}>x)=0,
which, along with (3.14), leads to
(3.16)P(∑i=1n‍Xie-rτi1{τi≤t}>x)≳∑i=1n‍P(Xie-rτi1{τi≤t}>x)
that holds uniformly for all t∈ΛT. On the other hand, we have that, for any fixed 0<v<1,
(3.17)P(∑i=1n‍Xie-rτi1{τi≥t}>x)≤P(⋃i=1n(Xie-rτi1{τi≤t}>vx))+P(∑i=1n‍Xie-rτi1{τi≤t}>x,⋂i=1n(Xie-rτi1{τi≤t}≤vx))=A4+A5.
For A4, it follows from F∈𝒟 that for any fixed ε>0, there exists a x1>0 such that, for all x≥x1 and all t∈ΛT, we have
(3.18)A4≤∑i=1n‍P(Xie-rτi1{τi≤t}>vx)~∑i=1n‍P(Xie-r(τi-1+θ*)1{τi-1+θ*≤t}>vx)=∑i=1n‍∫0t‍F¯(vxers)P(τi-1+θ*∈ds)≤(1+ε)(F*¯(v-1))-1∑i=1n‍∫0t‍F¯(xers)P(τi-1+θ*∈ds)=(1+ε)(F*¯(v-1))-1∑i=1n‍P(Xie-r(τi-1+θ*)1{τi-1+θ*≤t}>x)~(1+ε)(F*¯(v-1))-1∑i=1n‍P(Xie-rτi1{τi≤t}>x).
Hence, we can derive by the arbitrariness of ε>0 and 0<v<1 that
(3.19)A4≲LF-1∑i=1n‍P(Xie-rτi1{τi≤t}>x)
holds uniformly for all t∈ΛT, where LF-1 is of sense from Lemma 3.1(1). For A5, it holds uniformly for all t∈ΛT that
(3.20)A5=P(∑i=1n‍Xie-rτi1{τi≤t}>x,xn<max1≤k≤nXke-rτk1{τk≤t}≤vx)≤∑k=1n‍P(∑i=1,i≠kn‍Xie-rτi1{τi≤t}>(1-v)x,Xke-rτk1{τk≤t}>xn)≤∑k=1n‍∑i=1,i≠kn‍P(Xie-rτi1{τi≤t}>(1-v)xn,Xke-rτk1{τk≤t}>(1-v)xn)=o(1)∑i=1n‍∑i=1,i≠kn‍(P(Xie-rτi1{τi≤t}>(1-v)xn)+P(Xke-rτk1{τk≤t}>(1-v)xn))=o(1)∑i=1n‍P(Xie-rτi1{τi≤t}>x),
where in the second last step we used Lemma 3.3, and in the last step we used F∈𝒟 and Lemma 3.2(1). Hence, substituting (3.19) and (3.20) into (3.17), it holds uniformly for all t∈ΛT that
(3.21)P(∑i=1n‍Xie-rτi1{τi≥t}>x)≲LF-1∑i=1n‍P(Xie-rτi1{τi≤t}>x).
This, along with (3.16), proves the uniformity of (3.12) for all t∈ΛT.
Additionally, ifF∈𝒞, then LF=1, and thus we get the uniformity of (3.13) by (3.12).Lemma 3.5.
Under the conditions of Theorem2.6, relation (2.8) holds for every fixed t∈Λ. Additionally, if F∈𝒞, then relation (2.6) holds for every fixed t∈Λ.Proof.
Clearly, for every integeri≥2 and every fixed t∈Λ, Xie-rτi1{τi≤t}≤Xie-rτi-1, where Xi and e-rτi-1 are independent, note that, for any n≥1 and all t∈ΛT,
(3.22)P(∑i=n+1∞‍Xie-rτi1{τi≤t}>x)≤P(∑i=n+1∞‍Xie-rτi-1>x)=P(⋃i=n+1∞‍(Xie-rτi-1>x))+P(∑i=n+1∞‍Xie-rτi-1>x,⋂i=n+1∞‍(Xie-rτi-1≤x))≤∑i=n+1∞‍P(Xie-rτi-1>x)+P(∑i=n+1∞‍Xie-rτi-11{Xie-rτi-1≤x}>x)=A6+A7.
For A6, it follows by (3.1) that, for all x≥D1,
(3.23)A6=∑i=n+1∞‍∫01‍F¯(xy)dP(e-rτi-1≤y)≤C1-1F¯(x)∑i=n∞‍(Ee-rp1τ1)i.
Then for any given ε>0, there exists some positive integer n0 such that for all n≥n0,
(3.24)A6≤εF¯(x).
For A7, by Markov's inequality, we obtain that, for some p2>JF+,
(3.25)A7≤x-p2E(∑i=n+1∞‍Xie-rτi-11{Xie-rτi-1≤x})p2.
On the one hand when 0<JF+<1, applying the inequality |a+b|r≤|a|r+|b|r for 0<r<1 and any number a,b, we have
(3.26)A7≤x-p2∑i=n+1∞‍E(Xie-rτi-1)p21{Xie-rτi-1≤x}=x-p2∑i=n+1∞‍∫0∞‍∫0xers‍yp2e-rsp2dF(y)dP(τi-1≤s)≤x-p2∑i=n+1∞‍∫0∞‍∫0xers‍p2e-rsp2yp2-1F¯(y)dydP(τi-1≤s).
If xers<D2, then
(3.27)∫0xers‍p2e-rsp2yp2-1F¯(y)dy≤xp2≤xp2F¯(xers)F¯(D2).
If xers≥D2, then by (3.2) we get that
(3.28)∫0xers‍p2e-rsp2yp2-1F¯(y)dy=(∫0D2‍+∫D2xers‍)p2e-rsp2yp2-1F¯(y)dy≤C2F¯(D2)xp2F¯(xers)+C2∫D2xers‍p2xp2F¯(xers)y-1dy≤(C2F¯(D2)+C2p2ln(xersD2))xp2F¯(xers).
Hence, combining (3.27) and (3.28) and letting C=max{1/F¯(D2),C2/F¯(D2)+C2p2ln(xert/D2)} can show that
(3.29)∫0xers‍p2e-rsp2yp2-1F¯(y)dy≤Cxp2F¯(xers).
Substituting (3.29) into (3.26) and by (3.24), we deduce that, for all n≥n0,
(3.30)A7≤C∑i=n+1∞‍P(Xie-rτi-1>x)≤εF¯(x).
On the other hand when JF+>1, by Minkowski's inequality and along with the similar lines of the proof of the case when 0<JF+<1, we also attain that, for some constant C>0,
(3.31)A7≤x-p2(∑i=n+1∞‍(E(Xie-rτi-1)p21{Xie-rτi≤x})(1/p2))p2≤C(∑i=n+1∞‍(P(Xie-rτi-1>x))(1/p2))p2≤C(∑i=n+1∞‍(∫01‍F¯(xy)dP(e-rτi-1≤y))(1/p2))p2≤C(∑i=n∞‍(C1-1F¯(x)(Ee-rp1τ1)i)(1/p2))p2≤CC1-1F¯(x)(∑i=n∞‍(Ee-rτ1p1)(i/p2))p2,
where in the second last step we used (3.1). Therefore, for all n≥n0, it still holds that
(3.32)A7≤εF¯(x).
Consequently, from (3.22), (3.24), (3.30), and (3.32), we prove that, for some positive integer n0 and every fixed t∈Λ,
(3.33)P(∑i=n0+1∞‍Xie-rτi1{τi≤t}>x)≤εF¯(x),(3.34)∑i=n0+1∞‍P(Xie-rτi1{τi≤t}>x)≤∑i=n0+1∞‍P(Xie-rτi-1>x)≤εF¯(x).
Letn0 be fixed as above. Applying (3.12) in Lemma 3.4, (3.34), and Lemma 3.2 (2) in turn, we find that, for every fixed t∈Λ,
(3.35)P(Dr(t)>x)≥P(∑i=1n0‍Xie-rτi1{τi≤t}>x)~(∑i=1∞‍-∑i=n0+1∞‍)P(Xie-rτi1{τi≤t}>x)≥(1-ε)∑i=1∞‍P(Xie-rτi1{τi≤t}>x),
which, along with (1.7) and (2.7), proves that for every fixed t∈Λ,
(3.36)P(Dr(t)>x)≳(1-ε)∫0t‍F¯(xers)dλ~(t).
By contrast, for n0 as above and any fixed 0<v<1, we obtain from (3.12) and (3.33) that, for every fixed t∈Λ,
(3.37)P(Dr(t)>x)≤P(∑i=1n0‍Xie-rτi1{τi≤t}>(1-v)x)+P(∑i=n0+1∞‍Xie-rτi1{τi≤t}>vx)≲LF-1∑i=1n0‍P(Xie-rτi1{τi≤t}>(1-v)x)+εF¯(vx)=A8+A9.
For A8, arguing as the proof of (3.19) leads to
(3.38)A8≲LF-2∑i=1∞‍P(Xie-rτi1{τi≤t}>x).
For A9, by Lemmas 3.2 (2) and 3.2 (1) successively, we have
(3.39)A9=εP(X1e-rτ11{τ1≤t}>x).
From (3.37) to (3.39) and by the arbitrariness of ε>0, we obtain that, for every fixed t∈Λ,
(3.40)P(Dr(t)>x)≲LF-2∑i=1∞‍P(Xie-rτi1{τi≤t}>x)=LF-2∫0t‍F¯(xers)dλ~(t),
where in the last step we used (1.7) and (2.7) again. Hence, combining (3.36) and (3.40) can yield that relation (2.8) holds for every fixed t∈Λ.
Additionally, ifF∈𝒞, then LF=1, and so we attain by (2.8) that relation (2.6) holds for every fixed t∈Λ.


## 4. Proofs of Main Results

Proof of Theorem2.5.
From the proof of Lemma3.5, we can see that the relations (3.33)–(3.40) still hold uniformly for all t∈ΛT, and then we get the uniformity of (2.8) for all t∈ΛT immediately. As F∈𝒞, the uniformity of (2.6) over all t∈ΛT is clear.Proof of Theorem2.6.
According to the proof of Lemma 4.2 of Hao and Tang [7] (or see the proof of (4.3) of Tang [6]), we know that, for an arbitrarily fixed ε>0, there exists some T0∈Λ such that
(4.1)∫T0∞‍F¯(xers)dλ~(s)≤ε∫0T0‍F¯(xers)dλ~(s).
Combining with Theorem 2.5, it suffices to prove that relation (2.8) holds uniformly for all t∈(T0,∞]. On the one hand, by Lemma 3.5 with t=T0 and (4.1), it holds uniformly for all t∈(T0,∞] that
(4.2)P(Dr(t)>x)≥P(Dr(T0)>x)≳∫0T0‍F¯(xers)dλ~(s)≥(∫0t‍-∫T0∞‍)F¯(xers)dλ~(s)≥(1-ε)∫0t‍F¯(xers)dλ~(s).
On the other hand, by Lemma 3.5 and (4.1), it holds uniformly for all t∈(T0,∞] that
(4.3)P(Dr(t)>x)≲LF-2∫0∞‍F¯(xers)dλ~(s)≤LF-2(∫0t‍+∫T0∞‍)F¯(xers)dλ~(s)≤(1+ε)LF-2∫0t‍F¯(xers)dλ~(s).
By relations (4.2), (4.3), and the arbitrariness of ε>0, we get the uniformity of relation (2.8) for all t∈(T0,∞].
AsF∈𝒞, we have LF=1, and hence relation (2.6) holds uniformly for all t∈Λ.Proof of Theorem2.7.
Recalling the insurer's surplus process (1.2), we can attain its discounted value as
(4.4)U~r(t)=e-rtUr(t)=x+C~(t)-D(t),t≥0.
Following the definition (1.4) of the finite-time ruin probability, we have
(4.5)ψr(x,t)=P(U~r(s)<0for  some0<s≤t)=P(sup0<s≤t{Dr(s)-C~(s)}>x).
Then, it follows that
(4.6)ψr(x,t)≤P(Dr(t)>x),(4.7)ψr(x,t)=P(⋃0<s≤t‍{Dr(s)-C~(s)}>x)≥P(Dr(t)>x+C~(t)).
By Theorem 2.5 and (4.6), it holds uniformly for all t∈ΛT that
(4.8)ψr(x,t)≲LF-2∫0t‍F¯(xers)dλ~(s).
In the following, we establish the uniform asymptotic lower bound of ψr(x,t) for all t∈ΛT. Under the condition 1 of Theorem 2.7, we deduce from (4.7), Theorem 2.5, and F∈𝒟 that, for any fixed 0<w<1 and any given ε>0, there exists some x2>0 such that for all x≥x2 and uniformly for all t∈ΛT,
(4.9)ψr(x,t)≥P(Dr(t)>x+C~(T))=∫0∞‍P(Dr(t)>x+y)P(C~(T)∈dy)≳∫0∞‍∫0t‍F¯((x+y)ers)dλ~(s)P(C~(T)∈dy)≥∫0∞‍∫0t‍F¯((1+w)xers)dλ~(s)P(C~(T)∈dy)≥(1-ε)F¯*(1+w)∫0t‍F¯(xers)dλ~(s),
which, along with the arbitrariness of ε>0 and 0<w<1, can give that, uniformly for all t∈ΛT,
(4.10)ψr(x,t)≳LF∫0t‍F¯(xers)dλ~(s).
Under the condition 2 of Theorem 2.7, it holds from (4.7) that, for any fixed 0<δ<1 and all t∈ΛT,
(4.11)ψr(x,t)≥P(Dr(t)>(1+δ)x)-P(C~(T)>δx)=I1-I2.
For I1, using Theorem 2.5 and F∈𝒟, there exists some x3>0 such that for all x≥x3 and uniformly for all t∈ΛT,
(4.12)I1≳∫0t‍F¯((1+δ)xers)dλ~(s)≥(1-ε)F¯*(1+δ)∫0t‍F¯(xers)dλ~(s).
For I2, by the condition 2 of Theorem 2.7 and F∈𝒟, we have
(4.13)limsupx→∞I2F¯(x)=limsupx→∞I2F¯(δx)·F¯(δx)F¯(x)=0.
When t_∈Λ, by (4.13) and (3.2) in Lemma 3.1, there exists some x4>0 such that for all x≥max{D2,x4} and uniformly for all t∈ΛT,
(4.14)I2≤εF¯(x)≤Cε∫0t_‍F¯(xers)dλ~(s)≤Cε∫0t‍F¯(xers)dλ~(s),
where C=C2erp2t_/λ~(t_). When t_∉Λ, choose some 0<σ<1 such that t_+σ≤t, again by (4.13), (3.2), and arguing as (4.14), there exists some x4′>0 such that for all x≥max{D2,x4′} and uniformly for all t∈ΛT,
(4.15)I2≤εF¯(x)≤C′ε∫0t_+σ‍F¯(xers)dλ~(s)≤C′ε∫0t‍F¯(xers)dλ~(s),
where C′=C2erp2(t_+σ)/λ~(t_+σ). Hence, from (4.11) to (4.15) and by the the arbitrariness of ε>0 and 0<δ<1, we still obtain the uniformity of (4.10) for all t∈ΛT. This ends the proof for the uniformity of (2.10) over all t∈ΛT.
IfF∈𝒞, then LF=1. Therefore, we conclude from (2.10) that relation (2.12) uniformly holds for all t∈ΛT.Proof of Corollary2.8.
Clearly, by Theorem2.7, we know that relation (2.13) holds uniformly for all t∈ΛT. Note that for all finite t∈Λ,
(4.16)λ~(t)F¯(x+κλ~(t))≤κ-1∫xx+κλ~(t)‍F¯(y)dy≤λ~(t)F¯(x).
By F∈𝒟, it follows that, for any given ε>0 and any fixed 0<α<1, there exists some x5>0 such that for all x≥x5 and uniformly for all t∈ΛT,
(4.17)λ~(t)F¯(x+κλ~(t))≥λ~(t)F¯(x+κλ~(T))≥λ~(t)F¯((1+α)x)≥(1-ε)F¯*(1+α)λ~(t)F¯(x),
which, along with the arbitrariness of ε>0 and 0<α<1, yields that
(4.18)λ~(t)F¯(x+κλ~(t))≳LFλ~(t)F¯(x)
holds uniformly for all t∈ΛT. Thus, we obtain from (4.16) and (4.18) that, uniformly for all t∈ΛT,
(4.19)LFλ~(t)F¯(x)≲κ-1∫xx+κλ~(t)‍F¯(y)dy≲λ~(t)F¯(x).
So, combining (2.13) and (4.19) leads to the uniformity of (2.14) for all t∈ΛT.
IfF∈𝒞, then LF=1, and hence the uniformity of (2.15) for all t∈ΛT is proved immediately.Proof of Theorem2.9.
Applying (4.6) and Theorem 2.6, relation (4.8) still holds uniformly for all t∈Λ. On the other hand, by Theorem 2.7, we also attain that relation (4.10) holds uniformly for all t∈ΛT under conditions 1 and 2 of Theorem 2.9. Hence, we only need to show the uniformity of (4.10) for all t∈(T,∞]. Under the condition 1 of Theorem 2.9, similarly to the proof of (4.9), we prove that, uniformly for all t∈(T,∞],
(4.20)ψr(x,t)≥P(Dr(T)>x+C~)=∫0∞‍P(Dr(T)>x+y)P(C~∈dy)≳∫0∞‍∫0T‍F¯((x+y)ers)dλ~(s)P(C~∈dy)≥∫0∞‍∫0T‍F¯((1+w)xers)dλ~(s)P(C~∈dy)≥(1-ε)F¯*(1+w)∫0T‍F¯(xers)dλ~(s)≥(1-2ε)F¯*(1+w)∫0t‍F¯(xers)dλ~(s),
where the last step is due to (4.1). Because ε>0 and 0<w<1 are arbitrary, the relation (4.10) holds uniformly for all t∈(T,∞].
Under the condition 2 of Theorem2.9, we derive from (4.7) that, for the fixed 0<δ<1 as above and all t∈(T,∞],
(4.21)ψr(x,t)≥P(Dr(T)>x+C~)≥P(Dr(T)>(1+δ)x)-P(C~>δx)=I3-I4.
For I3, by Theorem 2.6 with t=T and the similar proof of (4.12), it holds uniformly for all t∈(T,∞] that
(4.22)I3~∫0T‍F¯((1+δ)xers)dλ~(s)≥(1-ε)F¯*(1+δ)∫0T‍F¯(xers)dλ~(s)≥(1-2ε)F¯*(1+δ)∫0t‍F¯(xers)dλ~(s),
where the last step is also due to (4.1). For I4, by condition 2 of Theorem 2.9 and F∈𝒟, we get that, for all t∈(T,∞],
(4.23)limsupx→∞I4∫0t‍F¯(xers)dλ~(s)≤limsupx→∞I4∫0T‍F¯(xers)dλ~(s)≤limsupx→∞I4F¯(δx)·F¯(δx)F¯(xerT)λ~(T)=0,
which implies that, for all t∈(T,∞] and all large x,
(4.24)I4≤ε∫0t‍F¯(xers)dλ~(s).
Consequently, using (4.21)–(4.24) and by the arbitrariness of ε>0 and 0<δ<1, we can obtain the uniformity of (4.10) for all t∈(T,∞], and then we prove the uniformity of (2.10) over all t∈Λ.
FromF∈𝒞 and (2.10), it is easy that relation (2.12) holds uniformly for all t∈ΛT.



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*Source: 186348-2012-09-20.xml*