VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA30 10.15559/15-VMSTA30 Research Article A Lundberg-type inequality for an inhomogeneous renewal risk model AndrulytėIeva Marijai.m.andrulyte@gmail.com BernackaitėEmilijaemilija.bernackaite@mif.vu.lt KievinaitėDominykad.kievinaite@gmail.com ŠiaulysJonasjonas.siaulys@mif.vu.lt∗ Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania Corresponding author. 2015 317201522173184 1962015 2572015 2672015 © 2015 The Author(s). Published by VTeX2015 Open access article under the CC BY license.

We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal risk model. We consider the model with independent, but not necessarily identically distributed, claim sizes and the interoccurrence times. In order to prove the main theorem, we first formulate and prove an auxiliary lemma on large values of a sum of random variables asymptotically drifted in the negative direction.

 Inhomogeneous model renewal model Lundberg-type inequality exponential bound ruin probability 91B30 60G50 Introduction

The classical risk model and the renewal risk model are two models that are traditionally used to describe the nonlife insurance business. The classical risk model was introduced by Lundberg and Cramér about a century ago (see [8, 14, 15] for the source papers and [18] for the historical environment). In this risk model, it is assumed that interarrival times are identically distributed, exponential, and independent random variables. In 1957, the Danish mathematician E. Sparre Andersen proposed the renewal risk model to describe the surplus process of the insurance company. In the renewal risk model, the claim sizes and the interarrival times are independent, identically distributed, nonnegative random variables (see [2] for the source paper and [22] for additional details). In this paper, we assume that interoccurrence times and claim sizes are nonnegative random variables (r.v.s) that are not necessarily identically distributed. We call such a model the inhomogeneous model and present its exact definition. It is evident that the inhomogeneous renewal risk model reflects better the real insurance activities in comparison with the classical risk model or with the renewal (homogeneous) risk model.

We say that the insurer’s surplus U(t) varies according to the inhomogeneous renewal risk model if U(t)=U(ω,t)=x+ct− ∑i=1Θ(t)Zi for all t⩾0 . Here:

x⩾0 is the initial reserve;

claim sizes {Z1,Z2,…} form a sequence of independent (not necessarily identically distributed) nonnegative r.v.s;

c>0 0$]]> is the constant premium rate;

Θ(t)=∑n=1∞1{Tn⩽t}=sup{n⩾0:Tn⩽t} is the number of claims in the interval [0,t] , where T0=0 , Tn=θ1+θ2+⋯+θn , n⩾1 , and the interarrival times {θ1,θ2,…} are independent (not necessarily identically distributed), nonnegative, and nondegenerate at zero r.v.s;

the sequences {Z1,Z2,…} and {θ1,θ2,…} are mutually independent.

A typical path of the surplus process of an insurance company is shown in Fig. 1.

Behavior of the surplus process.

If all claim sizes {Z1,Z2,…} and all interarrival times {θ1,θ2,…} are identically distributed, then the inhomogeneous renewal risk model becomes the homogeneous renewal risk model.

The time of ruin and the ruin probability are the main critical characteristics of any risk model. Let B denote the event of ruin. We suppose that B=⋃t⩾0{ω:U(ω,t)<0}=⋃t⩾0{ω:x+ct− ∑i=1Θ(t)Zi<0}, that is, that ruin occurs if at some time t⩾0 the surplus of the insurance company becomes negative or, in other words, the insurer becomes unable to pay all the claims. The first time τ when the surplus drops to a level less than zero is called the time of ruin, that is, τ is the extended r.v. for which τ=τ(ω)=inf{t⩾0:U(ω,t)<0}ifω∈B,∞ifω∉B.

The ruin probability ψ is defined by the equality ψ(x)=P(B)=P(τ=∞). Usually, we suppose that the main parameter of the ruin probability is the initial reserve x, though actually the ruin probability, together with time of ruin, depends on all components of the renewal risk model.

All trajectories of the process U(t) are increasing functions between times Tn and Tn+1 for all n=0,1,2,…  . Therefore, the random variables U(θ1+θ2+⋯+θn) , n⩾1 , are the local minimums of the trajectories. Consequently, we can express the ruin probability in the following way (for details, see [9] or [16]): ψ(x)=P(infn∈NU(θ1+θ2+⋯+θn)<0)=P(infn∈N{x+c(θ1+θ2+⋯+θn)− ∑i=1Θ(θ1+⋯+θn)Zi}<0)=P(infn∈N{x− ∑i=1n(Zi−cθi)}<0)=P(supn∈N{ ∑i=1n(Zi−cθi)}>x). x\Bigg).\end{array}\]]]>

Further, in this paper, we restrict our study to the so-called Lundberg-type inequality. An exponential bound for the ruin probability is usually called a Lundberg-type inequality. We further give the well-known exponential bound for ψ(x) in homogeneous renewal risk model (see, for instance, Chapters “Lundberg Inequality for Ruin Probability”, “Collective Risk Theory”, “Adjustment Coefficient,” or “Cramer–Lundberg Asymptotics” in [21]).

Let the net profit condition EZ1−cEθ1<0 hold, and let EehZ1<∞ for some h>0 0$]]> in the homogeneous renewal risk model. Then, there is a number H>0 0$]]> such that ψ(x)⩽e−Hx for all x⩾0 . If EeR(Z1−cθ1)=1 for some positive R, then we can take H=R in (1).

There exist a lot of different proofs of this theorem. The main ways to prove inequality (1) are described in Chapter “Lundberg Inequality for Ruin Probability” of [21]. Details of some existing proofs were given, for instance, by Asmussen and Albrecher [3], Embrechts, Klüppelberg, and Mikosch [9], Embrechts and Veraverbeke [10], Gerber [11], and Mikosch [16]. We note only that the bound (1) can be proved using the exponential tail bound of Sgibnev [19] and the inequality ψ(0)<1 .

The following theorem is the main statement of the paper.

Let the claim sizes {Z1,Z2,…} and the interarrival times {θ1,θ2,…} form an inhomogeneous renewal risk model described in Definition 1. Further, let the following three conditions be satisfied: (C1)supi∈NEeγZi<∞for someγ>0,(C2)limu→∞supi∈NE(θi1{θi>u})=0,(C3)lim supn→∞1n ∑i=1n(EZi−cEθi)<0. 0,\\{} \displaystyle (\mathcal{C}2)\hspace{2.84544pt}& \displaystyle \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(\theta _{i}\mathbb{1}_{\{\theta _{i}>u\}}\big)=0,\\{} \displaystyle (\mathcal{C}3)\hspace{2.84544pt}& \displaystyle \underset{n\to \infty }{\limsup }\frac{1}{n}{\sum \limits_{i=1}^{n}}(\hspace{0.1667em}\mathbb{E}Z_{i}-c\mathbb{E}\theta _{i})<0.\end{array}\]]]> Then, there are constants c1>0 0$]]> and c2⩾0 such that ψ(x)⩽e−c1x for all x⩾c2 .

The inhomogeneous renewal risk model differs from the homogeneous one because the independence and/or homogeneous distribution of sequences of random variables {Z1,Z2,…} and/or {θ1,θ2,…} are no longer required. The changes depend on how the inhomogeneity in a particular model is understood. In Definition 1, we have chosen one of two possible directions used in numerous articles that deal with inhomogeneous renewal risk models. This is due to the fact that an inhomogeneity can be considered as the possibility to have either differently distributed or dependent r.v.s in the sequences.

The possibility to have differently distributed r.v.s was considered, for instance, in [5, 6, 12, 17]. In the first three works, the discrete-time inhomogeneous risk model was considered. In such a model, the interarrival times are fixed, and the claims {Z1,Z2,…} are independent, not necessary identically distributed, integer valued r.v.s. In [17], the authors consider the model where the interarrival times are identically distributed and have a particular distribution, whereas the claims are differently distributed with distributions belonging to a particular class. Bernackaitė and Šiaulys [4] deal with an inhomogeneous renewal risk model where the r.v.s {θ1,θ2,…} are not necessarily identically distributed, but the claim sizes {Z1,Z2,…} have a common distribution function. In this article, we consider a more general renewal risk model. In the main theorem, we assume that not only r.v.s {θ1,θ2,…} are not necessarily identically distributed, but also the same holds for the sequence of claim sizes {Z1,Z2,…} .

There is another approach to inhomogeneous renewal risk models, which implies the possibility to have dependence in sequences and mainly found in works by Chinese researchers. In this kind of models, the sequences {Z1,Z2,…} and {θ1,θ2,…} consist of identically distributed r.v.s, but there may be some kind of dependence between them. Results for such models can be found, for instance, in [7] and [23]. Another interpretation of dependence is also possible, where r.v.s in both sequences {Z1,Z2,…} and {θ1,θ2,…} still remain independent. Instead of that, the mutual independence of these two sequences is no longer required. The idea of this kind of dependence belongs to Albrecher and Teugels [1], and this encouraged Li, Tang, and Wu [13] to study renewal risk models having this dependence structure.

The rest of the paper consists of two sections. In Section 2, we formulate and prove an auxiliary lemma. The proof of the main theorem is presented in Section 3.

 Auxiliary lemma

In order to prove the main theorem, we need an auxiliary lemma. In Lemma 1, the conditions for r.v.s η1 , η2 , η3,… are taken from articles by Smith [20] and Bernackaitė and Šiaulys [4].

Let η1 , η2 , η3,… be independent r.v.s such that (C1∗)supi∈NEeδηi<∞for someδ>0,(C2∗)limu→∞supi∈NE(|ηi|1{ηi⩽−u})=0,(C3∗)lim supn→∞1n ∑i=1nEηi<0. 0,\\{} \displaystyle \big(\mathcal{C}{2}^{\ast }\big)\hspace{2.84544pt}& \displaystyle \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}\leqslant -u\}}\big)=0,\\{} \displaystyle \big(\mathcal{C}{3}^{\ast }\big)\hspace{2.84544pt}& \displaystyle \underset{n\to \infty }{\limsup }\frac{1}{n}{\sum \limits_{i=1}^{n}}\mathbb{E}\eta _{i}<0.\end{array}\]]]> Then, there are constants c3>0 0$]]> and c4>0 0$]]> such that P(supk⩾1 ∑i=1kηi>x)⩽c3e−c4x x\Bigg)\leqslant c_{3}{\mathrm{e}}^{-c_{4}x}\]]]> for all x⩾0 .

First, we observe that, for all x⩾0 , P(supk⩾1 ∑i=1kηi>x)=P( ⋃k=1∞{ ∑i=1kηi>x})⩽ ∑k=1∞P( ∑i=1kηi>x). x\Bigg)& \displaystyle =\mathbb{P}\Bigg({\bigcup \limits_{k=1}^{\infty }}\Bigg\{{\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg\}\Bigg)\\{} & \displaystyle \leqslant {\sum \limits_{k=1}^{\infty }}\mathbb{P}\Bigg({\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg).\end{array}\]]]>

By Chebyshev’s inequality, for all x⩾0 , 0<y⩽δ , and k∈N , we have P( ∑i=1kηi>x)=P(ey∑i=1kηi>eyx)⩽e−yx ∏i=1kEeyηi. x\Bigg)& \displaystyle =\mathbb{P}\big({\mathrm{e}}^{y{\textstyle\sum _{i=1}^{k}}\eta _{i}}>{\mathrm{e}}^{yx}\big)\\{} & \displaystyle \leqslant {\mathrm{e}}^{-yx}{\prod \limits_{i=1}^{k}}\mathbb{E}{\mathrm{e}}^{y\eta _{i}}.\end{array}\]]]>

Moreover, for all i∈N , 0<y⩽δ , and u>0 0$]]>, we have Eeyηi=1+yEηi+E(eyηi−1−yηi) and E(eyηi−1−yηi)=E((eyηi−1)1{ηi⩽−u})−yE(ηi1{ηi⩽−u})+E((eyηi−1−yηi)1{−u<ηi⩽0})+E((eyηi−1−yηi)1{ηi>0}). 0\}}\big).\end{array}\]]]>

In order to evaluate the absolute value of the remainder term in (4), we need the following inequalities: |ev−1|⩽|v|,v⩽0,|ev−v−1|⩽v22,v⩽0,|ev−v−1|⩽v22ev,v⩾0.

Using them, we get |E(eyηi−1−yηi)|⩽2yE(|ηi|1{ηi⩽−u})+y22E(ηi21{−u<ηi⩽0})+y22E(ηi2eyηi1{ηi>0})⩽2ysupi∈NE(|ηi|1{ηi⩽−u})+y2u22+y22supi∈NE(ηi2eyηi1{ηi>0}), 0\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant 2y\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}\leqslant -u\}}\big)+\frac{{y}^{2}{u}^{2}}{2}+\frac{{y}^{2}}{2}\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big({\eta _{i}^{2}}{\mathrm{e}}^{y\eta _{i}}\mathbb{1}_{\{\eta _{i}>0\}}\big),\end{array}\]]]> where i∈N , 0<y⩽δ , and u>0 0$]]>.

Since limv→∞eδv/2v2=∞, we have eδv/2⩾v2 for all v⩾c5 , where c5=c5(δ)>0 0$]]>.

Therefore, supi∈NE(ηi2eδηi/21{ηi>0})⩽supi∈NE(ηi2eδηi/21{0<ηi⩽c5})+supi∈NE(ηi2eδηi/21{ηi>c5})⩽(c52+1)supi∈NEeδηi<∞. 0\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant \underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big({\eta _{i}^{2}}{\mathrm{e}}^{\delta \eta _{i}/2}\mathbb{1}_{\{0<\eta _{i}\leqslant c_{5}\}}\big)+\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big({\eta _{i}^{2}}{\mathrm{e}}^{\delta \eta _{i}/2}\mathbb{1}_{\{\eta _{i}>c_{5}\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant \big({c_{5}^{2}}+1\big)\underset{i\in \mathbb{N}}{\sup }\mathbb{E}{\mathrm{e}}^{\delta \eta _{i}}<\infty .\end{array}\]]]>

Choosing u=1y4 in (5) and using (6), we get |E(eyηi−1−yηi)|⩽2ysupi∈NE(|ηi|1{ηi⩽−1y4})+y322+y22supi∈NE(ηi2eyηi1{ηi>0})⩽y(2supi∈NE(|ηi|1{ηi⩽−1y4})+y122+y2(c52+1)supi∈NEeδηi)=:yα(y), 0\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant y\bigg(2\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}\leqslant -\frac{1}{\sqrt[4]{y}}\}}\big)+\frac{{y}^{\frac{1}{2}}}{2}+\frac{y}{2}\big({c_{5}^{2}}+1\big)\underset{i\in \mathbb{N}}{\sup }\mathbb{E}{\mathrm{e}}^{\delta \eta _{i}}\bigg)\\{} & \displaystyle \hspace{1em}=:y\alpha (y),\end{array}\]]]> where i∈N , y∈(0,δ/2] , c5=c5(δ) , and α(y)=2supi∈NE(|ηi|1{ηi⩽−1y4})+y122+y2(c52+1)supi∈NEeδηi.

Conditions ( C1∗ ) and ( C2∗ ) imply that α(y)↓0 as y→0 .

For an arbitrary positive v, we have supi∈NE(|ηi|1{ηi<0})=supi∈NE(|ηi|1{−v<ηi<0}+|ηi|1{ηi⩽−v})⩽v+supi∈NE(|ηi|1{ηi⩽−v}).

So, condition ( C2∗ ) implies that supi∈NE(|ηi|1{ηi<0})<∞.

Denote yˆ=min{δ/2,1/(2supi∈NE(|ηi|1{ηi<0}))}.

If y∈(0,yˆ] , then y(Eηi+α(y))>yEηi=yE(ηi1{ηi⩾0}+ηi1{ηi<0})⩾yE(ηi1{ηi<0})⩾yˆinfi∈NE(ηi1{ηi<0})=−yˆsupi∈NE(|ηi|1{ηi<0})⩾−1/2 y\mathbb{E}\eta _{i}\\{} & \displaystyle =y\mathbb{E}\big(\eta _{i}\mathbb{1}_{\{\eta _{i}\geqslant 0\}}+\eta _{i}\mathbb{1}_{\{\eta _{i}<0\}}\big)\\{} & \displaystyle \geqslant y\mathbb{E}\big(\eta _{i}\mathbb{1}_{\{\eta _{i}<0\}}\big)\\{} & \displaystyle \geqslant \widehat{y}\underset{i\in \mathbb{N}}{\inf }\mathbb{E}\big(\eta _{i}\mathbb{1}_{\{\eta _{i}<0\}}\big)\\{} & \displaystyle =-\widehat{y}\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(|\eta _{i}|\mathbb{1}_{\{\eta _{i}<0\}}\big)\\{} & \displaystyle \geqslant -1/2\end{array}\]]]> for all i∈N .

Therefore, (3), (4), (7), and the well-known inequality ln(1+u)⩽u,u>−1, -1,\]]]> imply that P( ∑i=1kηi>x)⩽e−yx ∏i=1k(1+yEηi+E(eyηi−1−yηi))⩽e−yx ∏i=1k(1+y(Eηi+α(y)))=exp{−yx+ ∑i=1kln(1+y(Eηi+α(y)))}⩽exp{−yx+y ∑i=1kEηi+ykα(y)}, x\Bigg)& \displaystyle \leqslant {\mathrm{e}}^{-yx}{\prod \limits_{i=1}^{k}}\big(1+y\mathbb{E}\eta _{i}+\mathbb{E}\big({\mathrm{e}}^{y\eta _{i}}-1-y\eta _{i}\big)\big)\\{} & \displaystyle \leqslant {\mathrm{e}}^{-yx}{\prod \limits_{i=1}^{k}}\big(1+y\big(\mathbb{E}\eta _{i}+\alpha (y)\big)\big)\\{} & \displaystyle =\exp \Bigg\{-yx+{\sum \limits_{i=1}^{k}}\ln \big(1+y\big(\mathbb{E}\eta _{i}+\alpha (y)\big)\big)\Bigg\}\\{} & \displaystyle \leqslant \exp \Bigg\{-yx+y{\sum \limits_{i=1}^{k}}\mathbb{E}\eta _{i}+yk\alpha (y)\Bigg\},\end{array}\]]]> where k∈N , x⩾0 , and y∈(0,yˆ] .

By estimate (8) and condition ( C3∗ ) we can suppose that lim supn→∞1n ∑i=1nEηi=−c6 for some positive constant c6 . Then we have 1k ∑i=1kEηi⩽−c62 for k⩾M+1 with some M⩾1 . Moreover, there exists y∗∈(0,yˆ] such that α(y∗)⩽c6/4 since α(y)↓0 as y→0 .

Using results from (2), (3), and (9), we derive P(supk⩾1 ∑i=1kηi>x)⩽ ∑k=1MP( ∑i=1kηi>x)+ ∑k=M+1∞P( ∑i=1kηi>x)⩽ ∑k=1Me−y∗x ∏i=1kEey∗ηi+ ∑k=M+1∞P( ∑i=1kηi>x)⩽ ∑k=1Me−y∗x ∏i=1kEey∗ηi+ ∑k=M+1∞e−y∗x+y∗∑i=1kEηi+y∗kα(y∗)⩽e−y∗x( ∑k=1M ∏i=1kEey∗ηi+ ∑k=0∞e−ky∗c6/4)⩽e−y∗x( ∑k=1M ∏i=1kΔ+11−e−y∗c6/4)=e−y∗x(Δ(ΔM−1)Δ−1+ey∗c6/4ey∗c6/4−1)=:c3e−c4x, x\Bigg)\\{} & \displaystyle \hspace{1em}\leqslant {\sum \limits_{k=1}^{M}}\mathbb{P}\Bigg({\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)+{\sum \limits_{k=M+1}^{\infty }}\mathbb{P}\Bigg({\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)\\{} & \displaystyle \hspace{1em}\leqslant {\sum \limits_{k=1}^{M}}{\mathrm{e}}^{-{y}^{\ast }x}{\prod \limits_{i=1}^{k}}\mathbb{E}{\mathrm{e}}^{{y}^{\ast }\eta _{i}}+{\sum \limits_{k=M+1}^{\infty }}\mathbb{P}\Bigg({\sum \limits_{i=1}^{k}}\eta _{i}>x\Bigg)\\{} & \displaystyle \hspace{1em}\leqslant {\sum \limits_{k=1}^{M}}{\mathrm{e}}^{-{y}^{\ast }x}{\prod \limits_{i=1}^{k}}\mathbb{E}{\mathrm{e}}^{{y}^{\ast }\eta _{i}}+{\sum \limits_{k=M+1}^{\infty }}{\mathrm{e}}^{-{y}^{\ast }x+{y}^{\ast }{\textstyle\sum _{i=1}^{k}}\mathbb{E}\eta _{i}+{y}^{\ast }k\alpha ({y}^{\ast })}\\{} & \displaystyle \hspace{1em}\leqslant {\mathrm{e}}^{-{y}^{\ast }x}\Bigg({\sum \limits_{k=1}^{M}}{\prod \limits_{i=1}^{k}}\mathbb{E}{\mathrm{e}}^{{y}^{\ast }\eta _{i}}+{\sum \limits_{k=0}^{\infty }}{\mathrm{e}}^{-k{y}^{\ast }c_{6}/4}\Bigg)\\{} & \displaystyle \hspace{1em}\leqslant {\mathrm{e}}^{-{y}^{\ast }x}\Bigg({\sum \limits_{k=1}^{M}}{\prod \limits_{i=1}^{k}}\varDelta +\frac{1}{1-{\mathrm{e}}^{-{y}^{\ast }c_{6}/4}}\Bigg)\\{} & \displaystyle \hspace{1em}={\mathrm{e}}^{-{y}^{\ast }x}\bigg(\frac{\varDelta ({\varDelta }^{M}-1)}{\varDelta -1}+\frac{{\mathrm{e}}^{{y}^{\ast }c_{6}/4}}{{\mathrm{e}}^{{y}^{\ast }c_{6}/4}-1}\bigg)=:c_{3}{\mathrm{e}}^{-c_{4}x},\end{array}\]]]> where x⩾0,Δ=1+supi∈NEeδηi,c3=Δ(ΔM−1)Δ−1+ey∗c6/4ey∗c6/4−1,c4=y∗∈(0,yˆ] with M⩾1 , c6>0 0$]]>, and yˆ>0 0$]]> defined previously. The lemma is now proved.  □

 Proof of Theorem <xref rid="j_vmsta30_stat_003">2</xref>

In this section, we prove Theorem 2.

Since ψ(x)=P(supn⩾1{ ∑i=1n(Zi−cθi)}>x), x\Bigg),\]]]> the desired bound of Theorem 2 can be derived from auxiliary Lemma 1.

Namely, supposing that r.v.s Zi−cθi , i∈{1,2,…} , satisfy all conditions of Lemma 1, we get ψ(x)⩽c7e−c8x for all x⩾0 with some positive c7 , c8 independent of x.

Therefore, ψ(x)⩽c7e−c8x/2e−c8x/2⩽e−c8x/2, with x⩾max{0,(2lnc7)/c8} ,

Thus, it suffices to check all three assumptions in our lemma with random variables Zi−cθi,i∈N . The requirement ( C3∗ ) of Lemma 1 is evidently satisfied by condition ( C 3).

Next, it follows from ( C 1) that supi∈NEeγ(Zi−cθi)⩽supi∈NEeγZi<∞.

So, the requirement ( C1∗ ) holds too.

It remains to prove that limu→∞supi∈NE(|Zi−cθi|1{Zi−cθi⩽−u})=0.

To establish this, we use the inequality supi∈NE(|Zi−cθi|1{Zi−cθi⩽−u})⩽supi∈NE(Zi1{Zi−cθi⩽−u})+csupi∈NE(θi1{Zi−cθi⩽−u}).

Taking the limit as u→∞ in the first summand of the right side of inequality (11), we get limu→∞supi∈NE(Zi1{Zi−cθi⩽−u})⩽limu→∞supi∈NE(Zi1{Zi−cθi⩽−u}1{θi⩽u2c})+limu→∞supi∈NE(Zi1{Zi−cθi⩽−u}1{θi>u2c})⩽limu→∞supi∈NE(Zi1{Zi⩽−u/2})+limu→∞supi∈NE(Zi1{Zi−cθi⩽−u}1{θi>u2c})=limu→∞supi∈NE(Zi1{Zi−cθi⩽−u}1{θi>u2c})⩽limu→∞supi∈NE(Zi1{θi>u2c})=limu→∞supi∈NEZiP(θi>u2c)⩽supi∈NEZilimu→∞supi∈NP(θi>u2c). \frac{u}{2c}\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{Z_{i}\leqslant -u/2\}}\big)\\{} & \displaystyle \hspace{2em}+\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{Z_{i}-c\theta _{i}\leqslant -u\}}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}\big)\\{} & \displaystyle \hspace{1em}=\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{Z_{i}-c\theta _{i}\leqslant -u\}}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}\big)\\{} & \displaystyle \hspace{1em}\leqslant \underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(Z_{i}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}\big)\\{} & \displaystyle \hspace{1em}=\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}Z_{i}\mathbb{P}\Big(\theta _{i}>\frac{u}{2c}\Big)\\{} & \displaystyle \hspace{1em}\leqslant \underset{i\in \mathbb{N}}{\sup }\mathbb{E}Z_{i}\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{P}\Big(\theta _{i}>\frac{u}{2c}\Big).\end{array}\]]]>

Since x⩽eγx/γ , x⩾0 , condition ( C 1) implies that supi∈NEZi<∞.

In addition, limu→∞supi∈NP(θi>u2c)=limu→∞supi∈NE(θi1{θi>u2c}θi)⩽limu→∞2cusupi∈NE(θi1{θi>u2c})=0 \frac{u}{2c}\Big)& \displaystyle =\underset{u\to \infty }{\lim }\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\bigg(\frac{\theta _{i}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}}{\theta _{i}}\bigg)\\{} & \displaystyle \leqslant \underset{u\to \infty }{\lim }\frac{2c}{u}\underset{i\in \mathbb{N}}{\sup }\mathbb{E}\big(\theta _{i}\mathbb{1}_{\{\theta _{i}>\frac{u}{2c}\}}\big)=0\end{array}\]]]> by condition ( C 2).

Therefore, relations (12), (13), and (14) imply that limu→∞supi∈NE(Zi1{Zi−cθi⩽−u})=0.

Now taking the limit as u→∞ in the second summand of the right side of inequality (11), by condition ( C 2) we have limu→∞supi∈NE(θi1{Zi−cθi⩽−u})=limu→∞supi∈NE(θi1{θi⩾1c(Zi+u)})⩽limu→∞supi∈NE(θi1{θi⩾uc})=0. We now see that the desired equality (10) follows from (11), (15), and (16). This means that all requirements of Lemma 1 hold for r.v.s Zi−cθi , i∈N .  □

 Acknowledgments

We would like to thank the anonymous referee for extremely detailed and helpful comments.

 References Albrecher, H., Teugels, J.L.: Exponential behavior in the presence of dependence in risk theory. J. Appl. Probab. 43(1), 257–273 (2006). MR2225065. doi:10.1239/jap/1143936258 Andersen Sparre, E.: On the collective theory of risk in the case of contagion between the claims. In: Transactions XVth International Congress of Actuaries, vol. 2(6), pp. 219–229 (1957) Asmussen, S., Albrecher, H.: Ruin Probabilities. World Scientific Publishing (2010). MR2766220. doi:10.1142/9789814282536 Bernackaitė, E., Šiaulys, J.: The exponential moment tail of inhomogeneous renewal process. Stat. Probab. Lett. 97, 9–15 (2015). MR3299745. doi:10.1016/j.spl.2014.10.018 Bieliauskienė, E., Šiaulys, J.: Infinite time ruin probability in inhomogeneous claim case. Liet. Mat. Rink., LMD Darbai 51, 352–356 (2010) (ISSN 0132-2818, available at page www.mii.lt/LMR/) Blaževičius, K., Bieliauskienė, E., Šiaulys, J.: Finite-time ruin probability in the inhomogeneous claim case. Lith. Math. J. 50(3), 260–270 (2010). MR2719562. doi:10.1007/s10986-010-9084-2 Chen, Y., Ng, K.W.: The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims. Insur. Math. Econ. 40(3), 415–423 (2007). MR2310980. doi:10.1016/j.insmatheco.2006.06.004 Cramér, H.: Historical review of Filip Lundberg’s works on risk theory. Scand. Actuar. J. 1969, 6–12 (1969). MR0347028. doi:10.1080/03461238.1969.10404602 Embrechts, P., Klüppelberg, C., Mikosch, T.: Modeling Extremal Events. Springer (1997). MR1458613. doi:10.1007/978-3-642-33483-2 Embrechts, P., Veraverbeke, N.: Estimates for probability of ruin with special emphasis on the possibility of large claims. Insur. Math. Econ. 1(1), 55–72 (1982). MR0652832. doi:10.1016/0167-6687(82)90021-X Gerber, H.U.: Martingales in risk theory. Mitt., Schweiz. Ver. Versicher.math. 73, 205–216 (1973) Lefèvre, C., Picard, P.: A nonhomogeneous risk model for insurance. Comput. Math. Appl. 51, 325–334 (2006). MR2203083. doi:10.1016/j.camwa.2005.11.005 Li, J., Tang, Q., Wu, R.: Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model. Adv. Appl. Probab. 42(4), 1126–1146 (2010). MR2796679. doi:10.1239/aap/1293113154 Lundberg, F.: Approximerad framställning av sannolikhetsfunktionen. Återförsäkring av kollektivrisker. Acad. Afhaddling. Almqvist. och Wiksell, Uppsala (1903) Lundberg, F.: Some supplementary researches on the collective risk theory. Skand. Aktuarietidskr. 15, 137–158 (1932) Mikosch, T.: Non-life Insurance Mathematics. Springer (2009). MR2503328. doi:10.1007/978-3-540-88233-6 Raducan, A.M., Vernic, R., Zbaganu, G.: Recursive calculation of ruin probabilities at or before claim instants for non-identically distributed claims. ASTIN Bull. 45, 421–443 (2015). doi:10.1017/asb.2014.30 Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Process for Insurance and Finance. John Wiley & Sons Ltd (1998). MR1680267. doi:10.1002/9780470317044 Sgibnev, M.S.: Submultiplicative moments of the suppremum of a random walk with negative drift. Stat. Probab. Lett. 32, 377–383 (1997). MR1602211. doi:10.1016/S0167-7152(96)00097-1 Smith, W.L.: On the elementary renewal theorem for non-identically distributed variables. Pac. J. Math. 14(2), 673–699 (1964). MR0165598. doi:10.2140/pjm.1964.14.673 Teugels, J., Sund, B. (eds.): Enciklopedia of Actuarial Science. Wiley (2004) Thorin, O.: Some comments on the Sparre Andersen model in the risk theory. ASTIN Bull. 8(1), 104–125 (1974). MR0351020 Wang, K., Wang, Y., Gao, Q.: Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab. 15(1), 109–124 (2013). MR3030214. doi:10.1007/s11009-011-9226-y