ADS Advances in Decision Sciences 2090-3367 2090-3359 Hindawi Publishing Corporation 936525 10.1155/2012/936525 936525 Research Article Uniform Estimate of the Finite-Time Ruin Probability for All Times in a Generalized Compound Renewal Risk Model Gao Qingwu 1 Jin Na 2 Zheng Juan 3 Chan Raymond 1 School of Mathematics and Statistics Nanjing Audit University, Nanjing 211815 China nau.edu.cn 2 Investment Department, Nanjing Times Media Co., Ltd., Nanjing 210039 China 3 School of Mathematics and Statistics Zaozhuang University, Zaozhuang 277160 China uzz.edu.cn 2012 25 10 2012 2012 04 04 2012 23 09 2012 2012 Copyright © 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.

1. 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.

1.1. Risk Model

It is well known that the compound renewal risk model was first introduced by Tang et al. , and since then it has been extensively investigated by many researchers, for example, Aleškevičienė et al. , Zhang et al. , Lin and Shen , Yang et al. , and the references therein. In the paper, we consider a generalized compound renewal risk model which satisfies the following assumptions.

Assumption <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

The interarrival times {θi,i1} of successive accidents are nonnegative, identically distributed, but not necessarily independent r.v.s with finite mean λ-1.

Assumption <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

The claim sizes and their number caused by nth accident are {Xi(n),i1} and Nn, n1, respectively, where {Xi(n),i1,n1} are nonnegative and identically distributed r.v.s with common distribution F and finite mean μ, and {Xi(n),i1} are not necessarily independent r.v.s, but {Xi(n),i1} and {Xi(m),i1} are mutually independent for all nm, n,m1, while {Nn,n1} are independent, identically distributed (i.i.d.), and positive integer-valued r.v.s with common distribution G and finite mean ν.

Assumption <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M20"><mml:mrow><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

The sequences {θi,i1}, {Xi(n),i1,n1}, and {Nn,n1} are mutually independent.

Denote the arrival times of the nth accident by τn=i=1nθi, n1, which can form a nonstandard renewal counting process (1.1)N(t)=sup{n1,τnt},t0, with mean function λ(t)=EN(t). Hence the total claim amount at time τn and the total claim amount up to time t0 are, respectively, (1.2)SNn(n)=i=1NnXi(n),  S(t)=n=1N(t)SNn(n), and then the insurer’s surplus process is given by (1.3)R(t)=x+ct-S(t),t0, where x0 is the initial surplus and c>0 is the constant premium rate. The finite-time ruin probability within time t>0 is defined as (1.4)ψ(x,t)=P(inf0stR(s)<0R(0)=x). Clearly, the ruin can only arise at the times τn, 1nN(t), then (1.5)ψ(x,t)=P(max0kN(t)n=1k(SNn(n)-cθn)>x). Let τ be a nonnegative r.v., the random time ruin probability is (1.6)ψ(x,τ)=P(max0kN(τ)n=1k(SNn(n)-cθn)>x).

In order for the ultimate ruin not to be certain, we assume the safety loading condition holds, namely, (1.7)κ=cλ-1-νμ>0.

In the generalized compound renewal risk model above, if all the sequences {θi,i1}, {Xi(n),i1,n1}, and {Nn,n1} are i.i.d. r.v.s, then the model is reduced to the standard compound renewal risk model introduced by Tang et al. , if N1=N2==1, then the model is the renewal risk model, see Tang , Leipus and Šiaulys , Yang et al. , and Wang et al. , among others.

1.2. 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 two positive functions a(·) and b(·), we write a(x)b(x) if limsupa(x)/b(x)1, write a(x)b(x) if liminfa(x)/b(x)1, write a(x)~b(x) if both, write a(x)=o(b(x)) if lima(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 tΔ if (1.8)limxsuptΔ|a(x,t)b(x,t)-1|=0. 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, (1.9)JV+=-limylogV¯*(y)logy,  JV-=-limylogV¯*(y)logy, where V¯*(y)=liminfV¯(xy)/V¯(x) and V¯*(y)=limsupV¯(xy)/V¯(x).

Chistyakov  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 V𝒮, if (1.10)V*2¯(x)~2V¯(x), where V*2 denotes the 2-fold convolution of V. Clearly, if V𝒮 then V is long tailed, denoted by V and characterized by (1.11)V¯(x+y)~V¯(x),        y>0. One can easily see that a distribution V if and only if there exists a function f(·):[0,)[0,) such that (1.12)f(x),f(x)=o(x),  V¯(x±f(x))~V¯(x). Korshunov  introduced a subclass of the class 𝒮, the strongly subexponential class, denoted by 𝒮*. Say that a distribution V𝒮*, if 0V¯(y)dy< and (1.13)Vu*2¯(x)~2Vu¯(x) holds uniformly for u[1,), where Vu¯(x)=min{1,xx+uV¯(y)dy}1{x0}+1{x<0} with 1A an indicator function of set A. Feller  introduced another important class of heavy-tailed distributions, the dominant variation class, which is not mutually inclusive with the class . Say that a distribution V on [0,) belongs to the dominant variation class, denoted by V𝒟, if (1.14)V¯*(y)<,        y>0. Cline  introduced a slightly smaller class of 𝒟, the consistent variation class, denoted by 𝒞. Say that a distribution V𝒞 if (1.15)limy1V¯*(y)=1,or  equivalently,limy1V¯*(y)=1. Specially, the class 𝒞 covers a famous class , called the regular variation class. By definition, a distribution V-α, if there exists some α>0 such that (1.16)limV¯(xy)V¯(x)=y-α,y>0. It is well known that for the distributions with finite mean, the following inclusion relationships hold properly, namely, (1.17)𝒞𝒟𝒮*𝒮, see, for example, Cline and Samorodnitsky , Klüppelberg , Embrechts et al. , and Denisov et al. . For more details of heavy-tailed distributions and their applications to finance and insurance, the readers are referred to Bingham et al.  and Embrechts et al. .

1.3. 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.  as follows.

Definition 1.1.

Say that r.v.s {ξi,i1} are widely upper orthant dependent (WUOD), if for each n1, there exists some finite positive number gU(n) such that, for all xi(-,), 1in, (1.18)P(i=1n{ξi>xi})gU(n)i=1nP(ξi>xi). Say that r.v.s {ξi,i1} are widely lower orthant dependent (WLOD), if for each n1, there exists some finite positive number gL(n) such that, for all xi(-,), 1in, (1.19)P(i=1n{ξixi})gL(n)i=1nP(ξixi). Furthermore, {ξi,i1} 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 gU(n)1 or gL(n)1 for each n1 in Definition 1.1, then {Xi,i1} are negatively upper orthant dependent or negatively lower orthant dependent (NLOD), see Ebrahimi and Ghosh  or Block et al. ; if gU(n)=gL(n)M for some constant M>0 and each n1 such that the two inequalities in Definition 1.1 both hold, then {Xi,i1} are extended negatively dependent, see Liu  and Chen et al. . 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 ), 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. .

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

Proposition 1.2.

( 1 ) Let {ξi,i1} be WLOD (or WUOD). If {fi(·),i1} are nondecreasing, then {fi(ξi),i1} are still WLOD (or WUOD); if {fi(·),i1} are nonincreasing, then {fi(ξi),i1} are WUOD (or WLOD).

( 2 ) If {ξi,i1} are nonnegative and WUOD, then for each n1, (1.20)Ei=1nξigU(n)  i=1nEξi.

Particularly, if {ξi,i1} are WUOD, then for each n1 and any s>0, (1.21)Eexp{si=1nξi}gU(n)i=1nEexp{sξi}.

Following the wide dependence structures as above, we will consider a generalized compound renewal risk model satisfying Assumption H3 and the following specific assumptions.

Assumption <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M142"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>

The interarrival times {θi,i1} are nonnegative, identically distributed and WLOD r.v.s with finite mean λ-1.

Assumption <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M145"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>

The claim sizes caused by nth accident {Xi(n),i1} are nonnegative, identically distributed and WUOD r.v.s, and the other statements of Assumption H2 are still valid.

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.

2. 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(τ1t)>0}.

2.1. 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, Aleškevičienė et al.  considered the standard compound renewal risk model with condition (1.7) and showed that

if G𝒞, F¯(x)=o(G¯(x)) and Eθ1p< for some p>JG++1, then it holds uniformly for all tΛ that (2.1)ψ(x,t)~1κxx+κλ(t)G(sμ)ds;

if F𝒞, G¯(x)=o(F¯(x)) and Eθ1p< for some p>JF++1, then it holds uniformly for all tΛ that (2.2)ψ(x,t)~νκxx+κλ(t)F(s)ds;

if F-α, G¯(x)~cF¯(x) for some constant c>0, and Eθ1p< for some p>α+1, then it holds uniformly for all tΛ that (2.3)ψ(x,t)~ν+cμακxx+κλ(t)F(s)ds.

Recently, Zhang et al.  extended the results of Aleškevičienė et al.  to the case that the claim sizes {Xi(n),i1} 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, G𝒞, and Eθ1p< for some p>JG++1; (ii) for c>0, F𝒞, and Eθ1p< for some p>JF++1; then it holds uniformly for all tΛ that(2.4)ψ(x,t)~1κxx+κλ(t)(νF(s)+G¯(sμ))ds. Observe the results of Zhang et al.  especially extended case (iii) of Aleškevičienė et al. . Also, Lin and Shen  considered a generalized compound renewal risk model with {Xi(n),i1} satisfying one type of asymptotically quadrant subindependent structure and also obtained the same relations (2.2), (2.3), and (2.4) as that of Aleškevičienė et al. .

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

to cancel the moment condition on {θi,i1}, namely, Eθ1p< for some p>JG++1, and Eθ1p< for some p>JF++1;

to extend partially the class 𝒞 or to the class 𝒟;

to discuss the case when {Xi(n),i1} are WUOD and {θi,i1} are WLOD;

to drop the interrelationships between F¯ and G¯ and investigate the case when both F and G are heavy tailed.

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

2.2. Main Results

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

Condition 1.

The interarrival times {θi,i1} are NLOD r.v.s.

Condition 2.

The interarrival times {θi,i1} are WOD r.v.s and there exists a positive and nondecreasing function g(x) such that g(x), x-kg(x)- for some 0<k<1, Eθ1g(θ1)<, and max{gU(n),gL(n)}g(n) for all n1, where x-kg(x)- means that x1-kg(x1)Cx2-kg(x2) for all 0x1<x2< and some finite constant C>0.

Condition 3.

The interarrival times {θi,i1} are WOD r.v.s with Eθ1p< for some 2p< and there exists a constant α>0 such that limnmax{gU(n),gL(n)}n-α=0.

Condition 4.

The interarrival times {θi,i1} are WOD r.v.s with Eeβθ1< for some 0<β< and limnmax{gU(n),gL(n)}e-γn=0 for any γ>0.

The first main result of this paper is the following.

Theorem 2.1.

Consider the generalized compound renewal risk model with Assumptions H1*, H2*, and H3 and condition (1.7), there exists a finite constant α>0 such that (2.5)limngL(n)n-α=0,  limngU(n)e-γn=0,foranyγ>0. Meanwhile, let one of Conditions 14 hold, and one of Conditions 14 with {θi,i1} replaced by {Xi(n),i1} still holds.

If F¯(x)~cG¯(x) for some c[0,], additionally, for c=0 and G𝒞; for 0<c< and F𝒞; for c= and F𝒟, then relation (2.4) holds uniformly for all tΛ.

If F𝒟 and G𝒞, relation (2.4) holds uniformly for all tΛ.

Note that there do exist some WLOD r.v.s satisfying condition (2.5), see Wang et al. . In the second main result below, we discuss the random time ruin probability, which requires another assumption.

Assumption <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M241"><mml:mrow><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

Let τ be nonnegative r.v. and independent of the sequences {θi,i1}, {Xin,i1,n1}, and {Nn,n1}.

Define Δ={τ:P(τΛ)>0}.

Theorem 2.2.

Under conditions of Theorem 2.1 and Assumption H4, one has

if F¯(x)~cG¯(x) for some c[0,], additionally, for c=0 and G𝒞; for 0<c< and F𝒞; for c= and F𝒟, then it holds uniformly for all tΔ that (2.6)ψ(x,τ)~1κExx+κλ(τ)(νF(s)+G¯(sμ))ds;

if F𝒟 and G𝒞, then relation (2.6) still holds uniformly for all tΔ.

Remark 2.3.

According to the proofs below of Theorems 2.1 and 2.2, we can see that Conditions 14 are doing nothing more than making {θi,i1} and {Xin,i1} satisfy the strong law of large number, namely, (2.7)limkk-1i=1kθi=λ-1,limkk-1i=1kXi(n)=μa.s.. So, Conditions 14 in Theorems 2.1 and 2.2 can be replaced by (2.7).

3. 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 {ξi,1in} are n WUOD and nonnegative r.v.s with distributions Vi𝒟,1in, respectively, then for any fixed 0<ab<, (3.1)P(i=1nciξi>x)~i=1nP(ciξi>x) holds uniformly for all cn_=(c1,c2,,cn)[a,b]n.

Proof.

See Lemma 3.1 of Gao et al. .

Particularly, let c1=c2=cn=1 in Lemma 3.2, we have a lemma below.

Lemma 3.2.

If {ξi,1in} are n WUOD and nonnegative r.v.s with distributions Vi𝒟,1in, then (3.2)P(i=1nξi>x)~i=1nV¯i(x).

Lemma 3.3.

If {ξi,i1} are WUOD and real-valued r.v.s with common distribution V𝒟 and mean 0 and satisfying (3.3)limngU(n)e-γn=0forany  γ>0, then for any γ>0, there exists a constants C=C(γ) such that (3.4)P(i=1nξi>x)CnV¯(x) holds for all xγn and all n1.

Proof.

By Proposition 1.2 and along the same lines of the proof of Theorem 3.1 of Tang  with slight modifications, we can derive that, for some positive integer m, (3.5)supnm,xγnP(i=1nξi>x)nV¯(x)<. From Lemma 3.2, we have (3.6)sup1nm,xγnP(i=1nξi>x)nV¯(x)n=1msupxγnP(i=1nξi>x)nV¯(x)<. Combining (3.5) and (3.6), there exists a constant C(γ)>0 such that (3.4) holds for all xγn and all n1.

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

Lemma 3.4.

Let {ξi,i1} be a sequence of real-valued r.v.s with finite mean a>0 and satisfy one of the Conditions 14 with {θi,i1} replaced by {ξi,i1}. Then (3.7)limnn-1i=1nξi=aa.s..

Proof.

Follow Theorem 1 of Matula , Theorem 1.4, and the proofs of Theorems 1.1 and 1.2 of Wang and Cheng , respectively.

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

Lemma 3.5.

Let {ξi,i1} 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 14 with {θi,i1} replaced by {ξi,i1}, 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 {ξi,i1}. Assume that V¯(x)~cU¯(x) for some c[0,].

If 0c<, U𝒞, and the conditions of Lemma 3.4 are valid, then (3.8)P(i=1ηξi>x)~bV¯(x)+U¯(xa).

If c= and V𝒟, then relation (3.8) holds.

Let no assumption be made on the interrelationship between V¯ and U¯. If V𝒟, U𝒞, 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 m0>0 such that, for any fixed ε>0, it holds that (3.9)Eη1{η>m0}ε.

( 1 ) First consider the case that 0<c<. Clearly, U𝒞 implies V𝒞. For any x>0 and any δ(0,1), we have (3.10)P(i=1ηξi>x)=n=1P(i=1nξi>x)P(η=n)=(n=1m0+n=m0+1(1-δ)x/a+n>(1-δ)x/a)P(i=1nξi>x)P(η=n)=K1+K2+K3. For K1, by Lemma 3.2 it follows that (3.11)K1~V¯(x)n=1m0nP(η=n)bV¯(x). For K2, since n<(1-δ)x/a and V𝒞𝒟, we obtain by Lemma 3.3 that (3.12)P(i=1nξi>x)=P(i=1n(ξi-a)>x-na)P(i=1n(ξi-a)>δx)C(γ)nV¯(δx)C~(γ)nV¯(x), 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)K2~V¯(x)n=m0+1(1-δ)x/anP(η=n)V¯(x)Eη1{η>m0}εV¯(x). For K3, since δ(0,1) can be arbitrarily close to 0, we see by U𝒞 that (3.14)K3U¯((1-δ)xa)U¯(xa). Substituting (3.11)–(3.14) into (3.10) and considering the arbitrariness of ε>0, we derive that (3.15)P(i=1ηξi>x)bV¯(x)+U¯(xa). On the other hand, we note that (3.16)P(i=1ηξi>x)=n=1P(i=1nξi>x)P(η=n)(n=1m0+n>(1+δ)x/a)P(i=1nξi>x)P(η=n)=K1+K4. For K1, by Lemma 3.2 and (3.9), we get (3.17)K1~(b-Eη1{η>m0})V¯(x)(b-ε)V¯(x). For K4, by Lemma 3.4 we find that (3.18)limnP(i=1nξin-a>-aδ1+δ)=1, which, along with V𝒞 and the arbitrariness of δ(0,1), leads to (3.19)K4n>(1+δ)x/aP(i=1nξin-a>-aδ1+δ)P(η=n)U¯((1+δ)xa)U¯(xa). Hence, from (3.16)–(3.19) and the arbitrariness of ε>0, we obtain that (3.20)P(i=1ηξi>x)bV¯(x)+U¯(xa). So, combining (3.15) and (3.20) proves that (3.8) holds for 0<c<.

Next we turn to the case that c=0, namely, V¯(x)=o(U¯(x)). According to Lemma 4.4 of Fay¨ et al. , there exists a nondecreasing slowly varying function L(x) such that V¯(x)=o(U¯(x)/L(x)), which results in that for some x0>0, (3.21)V¯(x)U¯(x)/L(x)1,forallxx0. Define (3.22)V*¯(x)={1,if0x<x0,U¯(x)L(x),ifxx0,V*-1(y)=inf{x(-,):V*(x)y},0y1, where V*(x)=1-V*¯(x). Let (3.23)ξi*=V*-1(V(ξi)),i1. It is easy to verify that {ξi*,i1} are still WUOD and identically distributed r.v.s with common distribution V*𝒞. By the definition of V*¯, we know that ξiξi*, i1, and then aEξi*=a*<. Thus, K1+K2 in (3.10) is divided into three parts as (3.24)K1+K2(n=1m0+n=m0+1(1-δ)x/a*)P(i=1nξi*>x)P(η=n)+(1-δ)x/a*<n(1-δ)x/aP(i=1nξi>x)P(η=n)=K1+K2+K2′′. Clearly, {ξi*,i1} are such that the conditions of Lemmas 3.2 and 3.3 hold, then (3.11) and (3.13) can still hold with K1, K2, and {ξi,i1} replaced by K1, K2, and {ξi*,i1}. So, we deduce by V*𝒞𝒟 and V*¯(x)=o(U¯(x)) that (3.25)K1+K2(b+ε)·V*¯(x)V*¯(x/a)·V*¯(x/a)U¯(x/a)·U¯(xa)=o(U¯(xa)). For K2′′, it follows from Lemma 3.4 that (3.26)limnP(i=1nξin-a>aδ1-δ)=0. Then, by U𝒞𝒟 we have (3.27)K2′′(1-δ)x/a*<n(1-δ)x/aP(i=1nξin-a>aδ1-δ)P(η=n)=o(U¯((1-δ)xa*))=o(U¯(xa)). From (3.10), (3.14), and (3.24)–(3.27), we find that (3.28)P(i=1ηξi>x)U¯(xa). Again by (3.16) and (3.19), it is seen that (3.29)P(i=1ηξi>x)K4U¯(xa). Since U𝒞𝒟 and V¯(x)=o(U¯(x)), we get (3.30)V¯(x)=V¯(x)U¯(x)·U¯(x)U¯(x/a)·U¯(xa)=o(U¯(xa)). Consequently, we obtain by combining (3.28)–(3.30) that (3.8) holds for c=0.

( 2 ) Now we deal with the case that c=, namely, U¯(x)=o(V¯(x)). Apparently, when V𝒟, we can derive by Lemmas 3.2 and 3.3 that (3.11) and (3.13) still hold. As for K3, by U¯(x)=o(V¯(x)) and V𝒟, we know that (3.31)K3U¯((1-δ)x/a)V¯((1-δ)x/a)·V¯((1-δ)x/a)V¯(x)·V¯(x)=o(V¯(x)). Substituting (3.11), (3.13), and (3.31) into (3.10) implies that (3.32)P(i=1ηξi>x)bV¯(x). For V𝒟, by Lemma 3.2 we also get (3.17). As for K4, arguing as (3.19) and (3.31), we still have that (3.33)K4U¯((1+δ)xa)=o(V¯(x)). From (3.16), (3.17), and (3.33), we conclude that (3.34)P(i=1ηξi>x)bV¯(x). Similarly to the derivation of (3.30), by U¯(x)=o(V¯(x)) and V𝒟 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 V𝒟, then (3.11) and (3.13) hold. While from the proof of (1), we have (3.14) if U𝒞. 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 V𝒟. Again by the proof of (1), relation (3.19) also holds for U𝒞. 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 N1=N2==1.

Condition 5.

For α in (2.5), there exist t0Λ(0,) and f in (1.12) such that (f(x))α(P(θ1t0))f(x)=o(F¯(x)).

Lemma 3.6 (Corollary 2.1 of Wang et al. [<xref ref-type="bibr" rid="B26">9</xref>]).

Consider the compound renewal risk model with N1=N2==1 and c>λμ, in which {Xi,i1} are i.i.d. r.v.s with common distribution F, and {θi,i1} are WLOD r.v.s satisfying (2.5) and one of Conditions 14.

If F𝒮*, then for any t0Λ, it holds uniformly for all t(t0,) that (3.35)ψ(x,t)~λc-λμxx+μλ(t)F¯(s)ds.

Furthermore, if Condition 5 holds, then relation (3.35) still holds uniformly for all tΛ.

Lemma 3.7 (Corollary <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M432"><mml:mn>2.2</mml:mn><mml:mo> </mml:mo><mml:mo> </mml:mo></mml:math></inline-formula>of Wang et al. [<xref ref-type="bibr" rid="B26">9</xref>]).

Under conditions of Lemma 3.6 and assumption H4, one has

if F𝒮*, then (3.36)ψ(x,τ)~λc-λμExx+μλ(τ)F¯(s)ds holds uniformly for all τ{τ:P(τI{τt0}Λ)>0} for any t0Λ.

Additionally, if Condition 5 holds, then relation (3.36) still holds uniformly for all tΔ.

Now we prove the main results as follows.

Proof of Theorem <xref ref-type="statement" rid="thm2.1">2.1</xref>.

Clearly, if F𝒟 then F satisfies Condition 5. In fact, the assumption F𝒟 indicates that the tail of F behaves essentially like a power function, thus there exists q>0 such that xqF¯(x). Take f(x)=xp for any 0<p<1, which satisfies (1.12). So, for any s>0, e-sxp/F¯(x)0 and Condition 5 holds.

First, consider Theorem 2.1(1). For the case that 0c<, 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 Theorems  2.2(2) and (3) of Aleškevičienė et al. . For the case that c=, since F𝒟, by Lemma 3.5(2) we have that for any y>0, (3.37)limsupxP(SN>xy)P(SN>x)=limsupxF¯(xy)F¯(x)<,  limxP(SN>x+y)P(SN>x)=limxF¯(x+y)F¯(x)=1, which tell us that the distribution of SN belongs to the class 𝒟. Then, arguing as the proof of Theorem 2.2(1) of Aleškevičienė et al. , we also obtain relation (2.4).

Now consider Theorem 2.1(2). By Lemma 3.5(3), F𝒟 and G𝒞, it is also clear that the distribution of SN belongs to 𝒟. Hence, relation (2.4) still holds from (1.5) and Lemmas 3.5(3)and 3.6.

Proof of Theorem <xref ref-type="statement" rid="thm2.2">2.2</xref>.

By (1.6), the uniformity of (2.4) in Theorem 2.1, and the independence between τ and the risk system, we can get the proof of Theorem 2.2.

Acknowledgments

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).

Tang Q. Su C. Jiang T. Zhang J. Large deviations for heavy-tailed random sums in compound renewal model Statistics & Probability Letters 2001 52 1 91 100 10.1016/S0167-7152(00)00231-5 1820135 ZBL0977.60034 Aleškevičienė A. Leipus R. Šiaulys J. Tail behavior of random sums under consistent variation with applications to the compound renewal risk model Extremes 2008 11 3 261 279 10.1007/s10687-008-0057-3 2429907 Zhang J. Cheng F. Wang Y. Tail behavior of random sums of negatively associated increments Journal of Mathematical Analysis and Applications 2011 376 1 64 73 10.1016/j.jmaa.2010.10.001 2745388 ZBL1207.60036 Lin Z. Shen X. Approximation of the tail probability of dependent random sums under consistent variation and applications Methodology and Computing in Applied Probability. In press 10.1007/s11009-011-9232-0. Yang Wang K Y. Liu J. Asymptotics and uniform asymptotics for finite-time and infinite-time absolute ruin probabilities in a dependent compound renewal risk model Journal of Mathematical Analysis and Applications 2013 398 1 352 361 10.1016/j.jmaa.2012.08.060 Tang Q. Asymptotics for the finite time ruin probability in the renewal model with consistent variation Stochastic Models 2004 20 3 281 297 10.1081/STM-200025739 2082126 ZBL1130.60312 Leipus R. Šiaulys J. Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes Insurance: Mathematics & Economics 2007 40 3 498 508 10.1016/j.insmatheco.2006.07.006 2311546 ZBL1183.91073 Yang Y. Leipus R. Šiaulys J. Cang Y. Uniform estimates for the finite-time ruin probability in the dependent renewal risk model Journal of Mathematical Analysis and Applications 2011 383 1 215 225 10.1016/j.jmaa.2011.05.013 2812731 ZBL1229.91169 Wang Y. Cui Z. Wang K. Ma X. Uniform asymptotics of the finite-time ruin probability for all times Journal of Mathematical Analysis and Applications 2012 390 1 208 223 10.1016/j.jmaa.2012.01.025 2885767 ZBL1237.91139 Chistyakov V. P. A theorem on sums of independent positive random variables and its applications to branching processes Theory of Probability and Its Applications 1964 9 640 648 10.1137/1109088 Korshunov D. A. Large deviation probabilities for the maxima of sums of independent summands with a negative mean and a subexponential distribution Theory of Probability and Its Applications 2001 46 2 355 366 10.1137/S0040585X97979019 1968696 Feller W. One-sided analogues of Karamata's regular variation L'Enseignement Mathématique 1969 15 107 121 0254905 ZBL0177.08201 Cline D. B. H. Intermediate regular and Π variation Proceedings of the London Mathematical Society 1994 68 3 594 616 10.1112/plms/s3-68.3.594 1262310 ZBL0793.26004 Cline D. B. H. Samorodnitsky G. Subexponentiality of the product of independent random variables Stochastic Processes and their Applications 1994 49 1 75 98 10.1016/0304-4149(94)90113-9 1258283 ZBL0799.60015 Klüppelberg C. Subexponential distributions and integrated tails Journal of Applied Probability 1988 25 1 132 141 929511 10.2307/3214240 ZBL0651.60020 Embrechts P. Klüppelberg C. Mikosch T. Modelling Extremal Events for Insurance and Finance 1997 33 Berlin, Germany Springer xvi+645 Applications of Mathematics 1458613 Denisov D. Foss S. Korshunov D. Tail asymptotics for the supremum of a random walk when the mean is not finite Queueing Systems 2004 46 1-2 15 33 10.1023/B:QUES.0000021140.87161.9c 2072274 ZBL1056.90028 Bingham N. H. Goldie C. M. Teugels J. L. Regular Variation 1987 27 Cambridge, UK Cambridge University Press xx+491 Encyclopedia of Mathematics and its Applications 898871 Wang K. Wang Y. Gao Q. Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate Methodology and Computing in Applied Probability. In press 10.1007/s11009-011-9226-y Ebrahimi N. Ghosh M. Multivariate negative dependence Communications in Statistics A 1981 10 4 307 337 612400 10.1080/03610928108828041 ZBL0506.62034 Block H. W. Savits T. H. Shaked M. Some concepts of negative dependence The Annals of Probability 1982 10 3 765 772 659545 10.1214/aop/1176993784 ZBL0501.62037 Liu L. Precise large deviations for dependent random variables with heavy tails Statistics & Probability Letters 2009 79 9 1290 1298 10.1016/j.spl.2009.02.001 2519013 ZBL1163.60012 Chen Y. Yuen K. C. Ng K. W. Precise large deviations of random sums in presence of negative dependence and consistent variation Methodology and Computing in Applied Probability 2011 13 4 821 833 10.1007/s11009-010-9194-7 2851839 ZBL1242.60027 Joag-Dev K. Proschan F. Negative association of random variables, with applications The Annals of Statistics 1983 11 1 286 295 10.1214/aos/1176346079 684886 ZBL0508.62041 Gao Q. Gu P. Jin N. Asymptotic behavior of the finite-time ruin probability with constant interest force and WUOD heavy-tailed claims Asia-Pacific Journal of Risk and Insurance 2012 6 1 10.1515/2153-3792.1129 Tang Q. Insensitivity to negative dependence of the asymptotic behavior of precise large deviations Electronic Journal of Probability 2006 11 4 107 120 10.1214/EJP.v11-304 2217811 ZBL1109.60021 Wang Y. Cheng D. Basic renewal theorems for random walks with widely dependent increments Journal of Mathematical Analysis and Applications 2011 384 2 597 606 10.1016/j.jmaa.2011.06.010 2825210 ZBL1230.60095 Matuła P. A note on the almost sure convergence of sums of negatively dependent random variables Statistics & Probability Letters 1992 15 3 209 213 10.1016/0167-7152(92)90191-7 1190256 ZBL0925.60024 Faÿ G. González-Arévalo B. Mikosch T. Samorodnitsky G. Modeling teletraffic arrivals by a Poisson cluster process Queueing Systems. Theory and Applications 2006 54 2 121 140 10.1007/s11134-006-9348-z 2268057 ZBL1119.60075