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  for the source papers and  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  for the source paper and  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 t0 . Here:

x0 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=11{Tnt}=sup{n0:Tnt} is the number of claims in the interval [0,t] , where T0=0 , Tn=θ1+θ2++θn , n1 , 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=t0{ω:U(ω,t)<0}=t0{ω:x+ct i=1Θ(t)Zi<0}, that is, that ruin occurs if at some time t0 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{t0:U(ω,t)<0}ifωB,ifωB. 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) , n1 , are the local minimums of the trajectories. Consequently, we can express the ruin probability in the following way (for details, see  or ): ψ(x)=P(infnNU(θ1+θ2++θn)<0)=P(infnN{x+c(θ1+θ2++θn) i=1Θ(θ1++θn)Zi}<0)=P(infnN{x i=1n(Zicθi)}<0)=P(supnN{ i=1n(Zicθi)}>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 ). Let the net profit condition EZ1cEθ1<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)eHx for all x0 . If EeR(Z1cθ1)=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 . Details of some existing proofs were given, for instance, by Asmussen and Albrecher , Embrechts, Klüppelberg, and Mikosch , Embrechts and Veraverbeke , Gerber , and Mikosch . We note only that the bound (1) can be proved using the exponential tail bound of Sgibnev  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)supiNEeγZi<for someγ>0,(C2)limusupiNE(θi1{θi>u})=0,(C3)lim supn1n i=1n(EZicEθi)<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 c20 such that ψ(x)ec1x for all xc2 .

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 . 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 , 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  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  and . 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 , and this encouraged Li, Tang, and Wu  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  and Bernackaitė and Šiaulys .

Let η1 , η2 , η3, be independent r.v.s such that (C1)supiNEeδηi<for someδ>0,(C2)limusupiNE(|ηi|1{ηiu})=0,(C3)lim supn1n 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(supk1 i=1kηi>x)c3ec4x x\Bigg)\leqslant c_{3}{\mathrm{e}}^{-c_{4}x}\]]]> for all x0 .

First, we observe that, for all x0 , P(supk1 i=1kηi>x)=P( k=1{ i=1kηi>x}) k=1P( 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 x0 , 0<yδ , and kN , we have P( i=1kηi>x)=P(eyi=1kηi>eyx)eyx 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 iN , 0<yδ , and u>0 0$]]>, we have Eeyηi=1+yEηi+E(eyηi1yηi) and E(eyηi1yηi)=E((eyηi1)1{ηiu})yE(ηi1{ηiu})+E((eyηi1yηi)1{u<ηi0})+E((eyηi1yηi)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: |ev1||v|,v0,|evv1|v22,v0,|evv1|v22ev,v0. Using them, we get |E(eyηi1yηi)|2yE(|ηi|1{ηiu})+y22E(ηi21{u<ηi0})+y22E(ηi2eyηi1{ηi>0})2ysupiNE(|ηi|1{ηiu})+y2u22+y22supiNE(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 iN , 0<yδ , and u>0 0$]]>.

Since limveδv/2v2=, we have eδv/2v2 for all vc5 , where c5=c5(δ)>0 0$]]>. Therefore, supiNE(ηi2eδηi/21{ηi>0})supiNE(ηi2eδηi/21{0<ηic5})+supiNE(ηi2eδηi/21{ηi>c5})(c52+1)supiNEeδηi<. 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(eyηi1yηi)|2ysupiNE(|ηi|1{ηi1y4})+y322+y22supiNE(ηi2eyηi1{ηi>0})y(2supiNE(|ηi|1{ηi1y4})+y122+y2(c52+1)supiNEeδηi)=: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{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 iN , y(0,δ/2] , c5=c5(δ) , and α(y)=2supiNE(|ηi|1{ηi1y4})+y122+y2(c52+1)supiNEeδηi. Conditions ( C1 ) and ( C2 ) imply that α(y)0 as y0 . For an arbitrary positive v, we have supiNE(|ηi|1{ηi<0})=supiNE(|ηi|1{v<ηi<0}+|ηi|1{ηiv})v+supiNE(|ηi|1{ηiv}). So, condition ( C2 ) implies that supiNE(|ηi|1{ηi<0})<. Denote yˆ=min{δ/2,1/(2supiNE(|ηi|1{ηi<0}))}. If y(0,yˆ] , then y(Eηi+α(y))>yEηi=yE(ηi1{ηi0}+ηi1{ηi<0})yE(ηi1{ηi<0})yˆinfiNE(ηi1{ηi<0})=yˆsupiNE(|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 iN . Therefore, (3), (4), (7), and the well-known inequality ln(1+u)u,u>1, -1,\]]]> imply that P( i=1kηi>x)eyx i=1k(1+yEηi+E(eyηi1yηi))eyx 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 kN , x0 , and y(0,yˆ] . By estimate (8) and condition ( C3 ) we can suppose that lim supn1n i=1nEηi=c6 for some positive constant c6 . Then we have 1k i=1kEηic62 for kM+1 with some M1 . Moreover, there exists y(0,yˆ] such that α(y)c6/4 since α(y)0 as y0 . Using results from (2), (3), and (9), we derive P(supk1 i=1kηi>x) k=1MP( i=1kηi>x)+ k=M+1P( i=1kηi>x) k=1Meyx i=1kEeyηi+ k=M+1P( i=1kηi>x) k=1Meyx i=1kEeyηi+ k=M+1eyx+yi=1kEηi+ykα(y)eyx( k=1M i=1kEeyηi+ k=0ekyc6/4)eyx( k=1M i=1kΔ+11eyc6/4)=eyx(Δ(ΔM1)Δ1+eyc6/4eyc6/41)=:c3ec4x, 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 x0,Δ=1+supiNEeδηi,c3=Δ(ΔM1)Δ1+eyc6/4eyc6/41,c4=y(0,yˆ] with M1 , 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(supn1{ i=1n(Zicθi)}>x), x\Bigg),\]]]> the desired bound of Theorem 2 can be derived from auxiliary Lemma 1.

Namely, supposing that r.v.s Zicθi , i{1,2,} , satisfy all conditions of Lemma 1, we get ψ(x)c7ec8x for all x0 with some positive c7 , c8 independent of x.

Therefore, ψ(x)c7ec8x/2ec8x/2ec8x/2, with xmax{0,(2lnc7)/c8} ,

Thus, it suffices to check all three assumptions in our lemma with random variables Zicθi,iN . The requirement ( C3 ) of Lemma 1 is evidently satisfied by condition ( C 3).

Next, it follows from ( C 1) that supiNEeγ(Zicθi)supiNEeγZi<.

So, the requirement ( C1 ) holds too.

It remains to prove that limusupiNE(|Zicθi|1{Zicθiu})=0.

To establish this, we use the inequality supiNE(|Zicθi|1{Zicθiu})supiNE(Zi1{Zicθiu})+csupiNE(θi1{Zicθiu}).

Taking the limit as u in the first summand of the right side of inequality (11), we get limusupiNE(Zi1{Zicθiu})limusupiNE(Zi1{Zicθiu}1{θiu2c})+limusupiNE(Zi1{Zicθiu}1{θi>u2c})limusupiNE(Zi1{Ziu/2})+limusupiNE(Zi1{Zicθiu}1{θi>u2c})=limusupiNE(Zi1{Zicθiu}1{θi>u2c})limusupiNE(Zi1{θi>u2c})=limusupiNEZiP(θi>u2c)supiNEZilimusupiNP(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 xeγx/γ , x0 , condition ( C 1) implies that supiNEZi<.

In addition, limusupiNP(θi>u2c)=limusupiNE(θi1{θi>u2c}θi)limu2cusupiNE(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 limusupiNE(Zi1{Zicθiu})=0.

Now taking the limit as u in the second summand of the right side of inequality (11), by condition ( C 2) we have limusupiNE(θi1{Zicθiu})=limusupiNE(θi1{θi1c(Zi+u)})limusupiNE(θi1{θiuc})=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 Zicθi , iN .  □

Acknowledgments

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

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