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Journal of Applied Mathematics
Volume 2012, Article ID 436531, 12 pages
http://dx.doi.org/10.1155/2012/436531
Research Article

Extended Precise Large Deviations of Random Sums in the Presence of END Structure and Consistent Variation

1School of Mathematic Sciences, Anhui University, Hefei, Anhui 230601, China
2Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Received 6 November 2011; Accepted 30 December 2011

Academic Editor: Ying U. Hu

Copyright © 2012 Shijie Wang and Wensheng Wang. 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.

Abstract

The study of precise large deviations of random sums is an important topic in insurance and finance. In this paper, extended precise large deviations of random sums in the presence of END structure and consistent variation are investigated. The obtained results extend those of Chen and Zhang (2007) and Chen et al. (2011). As an application, precise large deviations of the prospective- loss process of a quasirenewal model are considered.

1. Introduction

In the risk theory, heavy-tailed distributions are often used to model large claims. They play a key role in some fields such as insurance, financial mathematics, and queueing theory. We say that a distribution function 𝐹 belongs to the class 𝒞 iflim𝑦1liminf𝑥𝐹(𝑥𝑦)𝐹(𝑥)=1orequivalentlylim𝑦1limsup𝑥𝐹(𝑥𝑦)𝐹(𝑥)=1.(1.1) Such a distribution function 𝐹 is usually said to have a consistently varying tail. The heavy-tailed subclass 𝒞 was also studied by Cline and Samorodnitsky [1] who called it “intermediate regular variation.” Another well-known class is called the dominated variation class (denoted by 𝒟). A distribution function 𝐹 supported on (,) is in 𝒟 if and only iflimsup𝑥𝐹(𝑥𝑦)𝐹(𝑥)<(1.2) for any 0<𝑦<1 (or equivalently for some 0<𝑦<1). For more details of other heavy-tailed subclasses (e.g., ,𝒮,, and so on) and their relations, we refer the reader to [2] or [3].

Throughout this paper, let {𝑋𝑘,𝑘=1,2,} be a sequence of real-valued random variables with common distribution function 1𝐹(𝑥)=𝐹(𝑥)𝒞 and finite mean 𝜇. Let {𝑁(𝑡),𝑡0} be a nonnegative integer valued counting process independent of {𝑋𝑘,𝑘=1,2,} with mean function 𝜆(𝑡)=𝐸𝑁(𝑡), which tends to infinity as 𝑡. In insurance and finance, {𝑋𝑘,𝑘=1,2,} and {𝑁(𝑡),𝑡0} always denote the claims and claim numbers respectively. Hence, randomly indexed sums (random sums), which denote the loss process of the insurer during the period [0,𝑡], can be written as𝑆𝑁(𝑡)=𝑁(𝑡)𝑘=1𝑋𝑘,𝑡0.(1.3)

Recently, for practical reasons, precise large deviations of random sums with heavy tails have received a remarkable amount of attention. The study of precise large deviations is mainly to describe the deviations of a random sequence or a stochastic process away from its mean. The mainstream research of precise large deviations of 𝑆𝑁(𝑡) focuses on the study of the asymptotic relation𝑃𝑆𝑁(𝑡)𝜇𝜆(𝑡)>𝑥𝜆(𝑡)𝐹(𝑥),(1.4) which holds uniformly for some 𝑥-region 𝒟(𝑡) as 𝑡. The study of precise large deviations of random sums was initiated by Klüppelberg and Mikosch [4], who presented several applications in insurance and finance. For some latest works, we refer the reader to [2, 3, 511], among others.

In this paper, we are interested in the deviations of random sums 𝑆𝑁(𝑡) away from 𝑚𝜆(𝑡) with any fixed real number 𝑚. We aim at proving the following asymptotic relation:𝑃𝑆𝑁(𝑡)𝑚𝜆(𝑡)>𝑥𝜆(𝑡)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))(1.5) and the uniformity of (1.5). That is to say, in which 𝑥-region 𝒟(𝑡) as 𝑡 (1.5) holds uniformly. It is interesting that (1.5) reduces to (1.4) with 𝑚 replaced by 𝜇. Hence, we call (1.5) the extended precise large-deviation probabilities. More interestingly, setting 𝑚=0 and replacing 𝑋𝑘 with 𝑋𝑘(1+𝛿)𝜇 in (1.5), where 𝛽 denotes the safety loading coefficient, now (1.5) reduces to precise large-deviation probabilities for prospective-loss process. About precise large deviations for prospective-loss process, we refer the reader to [5].

The basic assumption of this paper is that {𝑋𝑘,𝑘=1,2,} is extended negatively dependent (END). The END structure was firstly introduced by Liu [12].

Definition 1.1. One calls random variables {𝑋𝑘,𝑘=1,2,} END if there exists a constant 𝑀>0 such that 𝑃𝑋1𝑥1,,𝑋𝑛𝑥𝑛𝑀𝑛𝑖=1𝑃𝑋𝑖𝑥𝑖,𝑃𝑋(1.6)1>𝑥1,,𝑋𝑛>𝑥𝑛𝑀𝑛𝑖=1𝑃𝑋𝑖>𝑥𝑖(1.7) hold for each 𝑛=1,2,, and all 𝑥1,,𝑥𝑛.

Recall that {𝑋𝑘,𝑘=1,2,} are called ND if both (1.6) and (1.7) hold with 𝑀=1; they are called positively dependent (PD) if inequalities (1.6) and (1.7) hold both in the reverse direction with 𝑀=1. According to Liu’s [12] interpretation, an ND sequence must be an END sequence. On the other hand, for some PD sequences, it is possible to find a corresponding positive constant 𝑀 such that (1.6) and (1.7) hold. Therefore, the END structure is substantially more comprehensive than the ND structure in that it can reflect not only a negative dependence structure but also a positive one to some extent.

Under the assumption that {𝑋𝑘,𝑘=1,2,} is an ND sequence, Liu [6] and Chen and Zhang [7] investigated precise large deviations of random sums 𝑆𝑁(𝑡) of nonnegative random variables and real-valued random variables, respectively. For a slightly more general dependence of END structure, Chen et al. [11] obtained precise large deviations of random sums 𝑆𝑁(𝑡) of nonnegative random variables and random sums 𝑆𝑁𝑡,𝑐=𝑁(𝑡)𝑘=1(𝑋𝑘+𝑐) of real-valued random variables with mean zero centered by a constant 𝑐. Up to now, to the best of our knowledge, little is known about extended precise large deviations of random sums in the presence of END structure and heavy tails. Our obtained results extend those of Chen and Zhang [7] and Chen et al. [11].

The rest of this paper is organized as follows. Section 2 gives some preliminaries. Precise large deviations of random sums in the presence of END real-valued random variables are presented in Section 3. In Section 4 we consider precise large deviations of the prospective-loss process of a quasirenewal model as an application of our main results.

2. Preliminaries

Throughout this paper, by convention, we denote 𝑆𝑛=𝑛𝑘=1𝑋𝑘. For two positive infinitesimals 𝑓() and 𝑔() satisfying𝑎liminf𝑓()𝑔()limsup𝑓()𝑔()𝑏,(2.1) we write 𝑓()=𝑂(𝑔()) if 𝑏<; 𝑓()=𝑜(𝑔()) if 𝑏=0; 𝑓()𝑔() if 𝑏=1; 𝑓()𝑔() if 𝑎=1; 𝑓()𝑔() if both and write 𝑓()𝑔() if 0<liminf(𝑓()/𝑔())limsup(𝑓()/𝑔())<. For theoretical and practical reasons, we usually equip them with certain uniformity. For instance, for two positive bivariate functions 𝑓(𝑡,𝑥) and 𝑔(𝑡,𝑥), we say that 𝑓(𝑡,𝑥)𝑔(𝑡,𝑥) holds as 𝑡 uniformly for all 𝑥𝒟(𝑡)𝜙 in the sense thatlim𝑡sup𝑥𝒟(𝑡)||||𝑓(𝑡,𝑥)||||𝑔(𝑡,𝑥)1=0.(2.2) For a distribution, set𝐽+𝐹=inflog𝐹(𝑦)log𝑦,𝑦>1,(2.3) where 𝐹(𝑦)=liminf𝑥(𝐹(𝑥𝑦))/𝐹(𝑥)). In the terminology of Tang and Tsitsiashvili [13], 𝐽+𝐹 is called the upper Matuszewska index of 𝐹. Clearly, if 𝐹𝒞, then 𝐽+𝐹<. It holds for every 𝑝>𝐽+𝐹 that𝑥𝑝=𝑜𝐹(𝑥),𝑥.(2.4) Moreover, 𝐽+𝐹1 if the distribution 𝐹(𝑥)=𝐹(𝑥)1{𝑥0} has a finite mean. See [11].

Next we will need some lemmas in the proof of our theorems. From Lemma 2.3 of Chen et al. [11] with a slight modification, we have the following lemma.

Lemma 2.1. Let {𝑋𝑘,𝑘=1,2,} be a sequence of real-valued END random variables with common distribution function 𝐹. If 0<𝜇+=𝐸𝑋11{𝑋10}<, then, for every fixed 𝜈>0 and some 𝐶=𝐶(𝜈)>0, the inequality 𝑃𝑆𝑛>𝑥𝑛𝐹𝑥𝜈𝑛+𝐶𝑥𝜈(2.5) holds for all 𝑛=1,2,, and 𝑥>0.

Lemma 2.2 below is a reformulation of Theorem 2.1 of [12], which is one of the main results in [12].

Lemma 2.2. Let {𝑋𝑘,𝑘=1,2,} be a sequence of real-valued END random variables with common distribution function 𝐹(𝑥)𝒞 and finite mean 𝜇, satisfying 𝐹(𝑥)=𝑜||𝑋𝐹(𝑥)as𝑥,𝐸1||𝑟1{𝑋10}<forsome𝑟>1.(2.6) Then, for any fixed 𝛾>0, relation 𝑃𝑆𝑛𝑛𝜇>𝑥𝑛𝐹(𝑥),as𝑛,(2.7) holds uniformly for all 𝑥𝛾𝑛.

3. Main Results and Their Proofs

In this sequel, all limiting relationships, unless otherwise stated, are according to 𝑡. To state the main results, we need the following two basic assumptions on the counting process {𝑁(𝑡),𝑡0}.

Assumption 3.1. For any 𝛿>0 and some 𝑝>𝐽+𝐹, 𝐸𝑁𝑝(𝑡)1(𝑁(𝑡)>(1+𝛿)𝜆(𝑡))=𝑂(𝜆(𝑡)).(3.1)

Assumption 3.2. The relation 𝑃(𝑁(𝑡)(1𝛿)𝜆(𝑡))=𝑜𝜆(𝑡)𝐹(𝜆(𝑡))(3.2) holds for all 0<𝛿<1.

Remark 3.3. One can easily see that Assumption 3.1 or Assumption 3.2 implies that 𝑁(𝑡)𝜆(𝑡)𝑃1.(3.3) See [5, 11].

Theorem 3.4. Let {𝑋𝑘,𝑘=1,2,} be a sequence of END real-valued random variables with common distribution function 𝐹(𝑥)𝒞 having finite mean 𝜇0 and satisfying (2.6), and let {𝑁(𝑡),𝑡0} be a nonnegative integer-valued counting process independent of {𝑋𝑘,𝑘=1,2,} satisfying Assumption 3.1. Let 𝑚 be a real number; then, for any 𝛾>(𝜇𝑚)0, the relation (1.5) holds uniformly for 𝑥𝛾𝜆(𝑡).

Theorem 3.5. Let {𝑋𝑘,𝑘=1,2,} be a sequence of END real-valued random variables with common distribution function 𝐹(𝑥)𝒞 having finite mean 𝜇<0 and satisfying (2.6), and let {𝑁(𝑡),𝑡0} be a nonnegative integer valued counting process independent of {𝑋𝑘,𝑘=1,2,}. (i)Assume that {𝑁(𝑡),𝑡0} satisfies Assumption 3.1 and 𝑚 is a real number (regardless of 𝑚0 or 𝑚<0), then for any fixed 𝛾>𝑚0, the relation (1.5) holds uniformly for 𝑥𝛾𝜆(𝑡).(ii)Assume that {𝑁(𝑡),𝑡0} satisfies Assumption 3.2 and 𝑚 is a negative real number; then, for any fixed 𝛾(𝜇𝑚0,𝑚], the relation (1.5) holds uniformly for 𝑥𝛾𝜆(𝑡).

Remark 3.6. One can easily see that Theorem 3.4 extends Theorem 3.1 of [11] with 𝑚 replaced by 𝜇. On the other hand, replacing 𝑋𝑘 with 𝑋𝑘𝜇+𝑐, setting 𝑚=0, and noticing that 𝐸(𝑋𝑘𝜇+𝑐)=𝑐, (3.4) yields Theorem 4.1(i) of [11].

Remark 3.7. Under the conditions of Theorem 3.5, choosing 𝑚=𝜇, one can easily see that the relation (1.4) holds uniformly over the 𝑥-region 𝑥𝛾𝜆(𝑡) for arbitrarily fixed 𝛾>0. Hence, Theorem 3.5 extends Theorem 1.2 of [7].

Proof of Theorem 3.4. We use the commonly used method with some modifications to prove Theorem 3.4. The starting point is the following standard decomposition: 𝑃𝑆𝑁(𝑡)=𝑚𝜆(𝑡)>𝑥𝑛=1𝑃𝑆𝑛=𝑚𝜆(𝑡)>𝑥𝑃(𝑁(𝑡)=𝑛)𝑛<(1𝛿)𝜆(𝑡)+(1𝛿)𝜆(𝑡)𝑛(1+𝛿)𝜆(𝑡)+𝑛>(1+𝛿)𝜆(𝑡)𝑆×𝑃𝑛𝑚𝜆(𝑡)>𝑥𝑃(𝑁(𝑡)=𝑛)=𝐼1(𝑥,𝑡)+𝐼2(𝑥,𝑡)+𝐼3(𝑥,𝑡),(3.4) where we choose 0<𝛿<1 such that (𝛾+𝑚)/(1+𝛿)𝜇>0.
We first deal with 𝐼1(𝑥,𝑡). Note that 𝑥+𝑚𝜆(𝑡)𝑛𝜇((𝛾+𝑚)/(1𝛿)𝜇)𝑛. Thus, as 𝑡 and uniformly for 𝑥𝛾𝜆(𝑡), it follows from Lemma 2.2 that𝐼1(𝑥,𝑡)𝑛<(1𝛿)𝜆(𝑡)𝑛𝐹(𝑥+𝑚𝜆(𝑡)𝑛𝜇)𝑃(𝑁(𝑡)=𝑛)𝑛<(1𝛿)𝜆(𝑡)𝐹(𝑥+𝑚𝜆(𝑡)(1𝛿)𝜇𝜆(𝑡))𝑛𝑃(𝑁(𝑡)=𝑛)(1𝛿)𝜆(𝑡)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))𝑃(𝑁(𝑡)<(1𝛿)𝜆(𝑡))=𝑜𝜆(𝑡).𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))(3.5)
Next, for 𝐼2(𝑥,𝑡), noticing that 𝑥+𝑚𝜆(𝑡)𝑛𝜇((𝛾+𝑚)/(1+𝛿)𝜇)𝑛, as 𝑡 and uniformly for 𝑥𝛾𝜆(𝑡), Lemma 2.2 yields that𝐼2(𝑥,𝑡)(1𝛿)𝜆(𝑡)𝑛(1+𝛿)𝜆(𝑡)𝑛𝐹(𝑥+𝑚𝜆(𝑡)𝑛𝜇)𝑃(𝑁(𝑡)=𝑛)(1+𝛿)𝜆(𝑡)||||𝐹(𝑥+𝑚𝜆(𝑡)𝜇(1+𝛿)𝜆(𝑡))𝑃𝑁(𝑡)||||𝜆(𝑡)1<𝛿(1+𝛿)𝜆(𝑡)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡)𝛿𝜇𝜆(𝑡))(1+𝛿)𝜆(𝑡)𝐹1𝛿𝜇.𝛾+𝑚𝜇(𝑥+(𝑚𝜇)𝜆(𝑡))(3.6) On the other hand, 𝐼2(𝑥,𝑡)(1𝛿)𝜆(𝑡)𝑛(1+𝛿)𝜆(𝑡)𝑛𝐹(𝑥+𝑚𝜆(𝑡)𝑛𝜇)𝑃(𝑁(𝑡)=𝑛)(1𝛿)𝜆(𝑡)||||𝐹(𝑥+𝑚𝜆(𝑡)𝜇(1𝛿)𝜆(𝑡))𝑃𝑁(𝑡)||||𝜆(𝑡)1<𝛿(1𝛿)𝜆(𝑡)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡)+𝛿𝜇𝜆(𝑡))(1𝛿)𝜆(𝑡)𝐹1+𝛿𝜇.𝛾+𝑚𝜇(𝑥+(𝑚𝜇)𝜆(𝑡))(3.7)
Finally, to deal with 𝐼3(𝑥,𝑡), we formulate the remaining proof into two parts according to 𝑚0 and 𝑚<0. In the case of 𝑚0, setting 𝜈=𝑝 in Lemma 2.1 with 𝑝>𝐽+𝐹1, for sufficiently large 𝑡 and 𝑥𝛾𝜆(𝑡), there exists some constant 𝐶1>0 such that𝐼3(𝑥,𝑡)𝑛>(1+𝛿)𝜆(𝑡)𝑃𝑆𝑛>𝑥𝑃(𝑁(𝑡)=𝑛)𝑛>(1+𝛿)𝜆(𝑡)𝑛𝐹𝑥𝑝+𝐶1𝑛𝑥𝑝𝑃(𝑁(𝑡)=𝑛).(3.8) In the case of 𝑚<0, note that 1+𝑚/𝛾>0 since 𝛾>𝜇𝑚𝑚>0. Similar to (3.8), for sufficiently large 𝑡 and 𝑥𝛾𝜆(𝑡), there exists some constant 𝐶2>0 such that 𝐼3(𝑥,𝑡)𝑛>(1+𝛿)𝜆(𝑡)𝑃𝑆𝑛>(1+𝑚/𝛾)𝑥𝑃(𝑁(𝑡)=𝑛)𝑛>(1+𝛿)𝜆(𝑡)𝑛𝐹(1+𝑚/𝛾)𝑥𝑝+𝐶2𝑛(1+𝑚/𝛾)𝑥𝑝𝑃(𝑁(𝑡)=𝑛).(3.9) As a result, by (2.4), as 𝑡 and uniformly for 𝑥𝛾𝜆(𝑡), both (3.8) and (3.9) yield that 𝐼3(𝑥,𝑡)𝐹(𝑥)𝐸𝑁(𝑡)1{𝑁(𝑡)>(1+𝛿)𝜆(𝑡)}+𝑥𝑝𝐸𝑁𝑝(𝑡)1{𝑁(𝑡)>(1+𝛿)𝜆(𝑡)}𝑜(1)𝐹(𝑥)𝐸𝑁𝑝(𝑡)1{𝑁(𝑡)>(1+𝛿)𝜆(𝑡)}𝑜(1)𝜆(𝑡)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡)),(3.10) where in the last step, we used 𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))𝐹||||1+𝑚𝜇𝛾𝑥𝐹(𝑥).(3.11)
Substituting (3.5), (3.6), (3.7), and (3.10) into (3.4), one can see that relation (1.5) holds by the condition 𝐹𝒞 and the arbitrariness of 𝛿.

Proof of Theorem 3.5. (i) We also start with the decomposition (3.4).
For 𝐼1(𝑥,𝑡) and 𝐼2(𝑥,𝑡), note that 𝑥+𝑚𝜆(𝑡)𝑛𝜇((𝛾+𝑚)/(1+𝛿)𝜇)𝑛 since 𝑛(1+𝛿)𝜆(𝑡). Hence, mimicking the proof of Theorem 3.4, we obtain, as 𝑡 and uniformly for 𝑥𝛾𝜆(𝑡),𝐼1(𝑥,𝑡)𝑛<(1𝛿)𝜆(𝑡)𝑛𝐹(𝑥+𝑚𝜆(𝑡)𝑛𝜇)𝑃(𝑁(𝑡)=𝑛)𝐹(𝑥+𝑚𝜆(𝑡))𝑛<(1𝛿)𝜆(𝑡)𝑛𝑃(𝑁(𝑡)=𝑛)=𝑜𝜆(𝑡),𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))(3.12) where, in the last step, we used the relation 𝐹(𝑥+𝑚𝜆(𝑡))𝐹𝑚1+𝛾𝑥𝐹(𝑥),(3.13)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))𝐹||||1+𝑚𝜇𝛾𝑥𝐹(𝑥).(3.14)
Again, as 𝑡 and uniformly for 𝑥𝛾𝜆(𝑡),𝐼2(𝑥,𝑡)(1𝛿)𝜆(𝑡)𝑛(1+𝛿)𝜆(𝑡)𝑛𝐹(𝑥+𝑚𝜆(𝑡)𝑛𝜇)𝑃(𝑁(𝑡)=𝑛)(1+𝛿)𝜆(𝑡)||||𝐹(𝑥+𝑚𝜆(𝑡)𝜇(1𝛿)𝜆(𝑡))𝑃𝑁(𝑡)||||𝜆(𝑡)1<𝛿(1+𝛿)𝜆(𝑡)𝐹1+𝛿𝜇(,𝐼𝛾+𝑚𝜇𝑥+(𝑚𝜇)𝜆(𝑡))2(𝑥,𝑡)(1𝛿)𝜆(𝑡)𝐹1𝛿𝜇.𝛾+𝑚𝜇(𝑥+(𝑚𝜇)𝜆(𝑡))(3.15)
Finally, in 𝐼3(𝑥,𝑡), setting 𝜈=𝑝 in Lemma 2.1 with 𝑝>𝐽+𝐹1, by (2.4) and Assumption 3.1, as 𝑡 and uniformly for 𝑥𝛾𝜆(𝑡), there exists a constant 𝐶2>0 such that𝐼3(𝑥,𝑡)𝑛>(1+𝛿)𝜆(𝑡)𝑃𝑆𝑛>(1+𝑚/𝛾)𝑥𝑃(𝑁(𝑡)=𝑛)𝑛>(1+𝛿)𝜆(𝑡)𝑛𝐹(1+𝑚/𝛾)𝑥𝑝+𝐶2𝑛(1+𝑚/𝛾)𝑥𝑝𝑃(𝑁(𝑡)=𝑛)𝐹(𝑥)𝐸𝑁(𝑡)1{𝑁(𝑡)>(1+𝛿)𝜆(𝑡)}+𝑥𝑝𝐸𝑁𝑝(𝑡)1{𝑁(𝑡)>(1+𝛿)𝜆(𝑡)}𝑜(1)𝜆(𝑡)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡)),(3.16) where in the last step we also used (3.14). Combining (3.12), (3.15), and (3.16), relation (1.5) holds by the condition 𝐹𝒞 and the arbitrariness of 𝛿.
(ii) We also start with the representation (3.4) in which we choose 0<𝛿<1 such that (𝛾+𝑚)/(1𝛿)𝜇>0.
To deal with 𝐼1(𝑥,𝑡), arbitrarily choosing 𝛾1>𝑚, we split the 𝑥-region into two disjoint regions as[𝛾𝛾𝜆(𝑡),)=1𝜆(𝑡),𝛾𝜆(𝑡),𝛾1𝜆(𝑡).(3.17) For the first 𝑥-region 𝑥𝛾1𝜆(𝑡), noticing that 𝑥+𝑚𝜆(𝑡)𝑛𝜇>|𝜇|𝑛, by Lemma 2.2, it holds uniformly for all 𝑥𝛾1𝜆(𝑡) that 𝐼1(𝑥,𝑡)𝑛<(1𝛿)𝜆(𝑡)𝑛𝐹(𝑥+𝑚𝜆(𝑡)𝑛𝜇)𝑃(𝑁(𝑡)=𝑛)(1𝛿)𝜆(𝑡)𝐹(𝑥+𝑚𝜆(𝑡))𝑃(𝑁(𝑡)(1𝛿)𝜆(𝑡))(1𝛿)𝜆(𝑡)𝐹(𝑥)𝑃(𝑁(𝑡)(1𝛿)𝜆(𝑡))(1𝛿)𝜆(𝑡)𝜆𝐹(𝑥(𝑚𝜇)𝜆(𝑡))𝑃(𝑁(𝑡)(1𝛿)𝜆(𝑡))=𝑜(𝑡)𝐹,(𝑥+(𝑚𝜇)𝜆(𝑡))(3.18) where the second step and the before the last before the step can be verified, respectively, as 𝐹(𝑥+𝑚𝜆(𝑡))𝐹𝑚1+𝛾1𝑥𝐹(𝑥),𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))𝐹||||1+𝑚𝜇𝛾1𝑥𝐹(𝑥).(3.19) For the second 𝑥-region 𝛾𝜆(𝑡)𝑥<𝛾1𝜆(𝑡), note that 𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))𝐹||||1+𝑚𝜇𝛾𝑥𝐹(𝑥)𝐹𝛾1𝜆(𝑡)𝐹(𝜆(𝑡)).(3.20) Hence, by Assumption 3.2, we still obtain 𝐼1(𝑥,𝑡)𝑃(𝑁(𝑡)(1𝛿)𝜆(𝑡))=𝑜(1)𝜆(𝑡)𝐹(𝜆(𝑡))𝑜𝜆(𝑡)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))(3.21) uniformly for all 𝛾𝜆(𝑡)𝑥<𝛾1𝜆(𝑡). As a result, the relation 𝐼1(𝑥,𝑡)=𝑜𝜆(𝑡)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))(3.22) holds uniformly for all 𝑥𝛾𝜆(𝑡).
For 𝐼2(𝑥,𝑡), since 𝛾(𝜇𝑚0,𝑚], it holds that𝑥+𝑚𝜆(𝑡)𝑛𝜇(𝛾+𝑚)𝜆(𝑡)𝜇𝑛𝛾+𝑚1𝛿𝜇𝑛.(3.23) It follows from Lemma 2.2 that for all 𝑥𝛾𝜆(𝑡)𝐼2(𝑥,𝑡)(1𝛿)𝜆(𝑡)𝑛(1+𝛿)𝜆(𝑡)𝑛𝐹(𝑥+𝑚𝜆(𝑡)𝑛𝜇)𝑃(𝑁(𝑡)=𝑛)(1+𝛿)𝜆(𝑡)||||𝐹(𝑥+𝑚𝜆(𝑡)𝜇(1𝛿)𝜆(𝑡))𝑃𝑁(𝑡)||||𝜆(𝑡)1<𝛿(1+𝛿)𝜆(𝑡)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡)+𝛿𝜇𝜆(𝑡))(1+𝛿)𝜆(𝑡)𝐹1+𝛿𝜇.𝛾+𝑚𝜇(𝑥+(𝑚𝜇)𝜆(𝑡))(3.24) Symmetrically, 𝐼2(𝑥,𝑡)(1𝛿)𝜆(𝑡)𝐹1𝛿𝜇𝛾+𝑚𝜇(𝑥+(𝑚𝜇)𝜆(𝑡)).(3.25)
Finally, in 𝐼3(𝑥,𝑡), note that 𝑥+𝑚𝜆(𝑡)𝑛𝜇(𝛾+𝑚)𝜆(𝑡)𝜇𝑛((𝛾+𝑚)/(1+𝛿)𝜇)𝑛. Therefore, Lemma 2.2 implies, as 𝑡 and uniformly for 𝑥𝛾𝜆(𝑡), that𝐼3(𝑥,𝑡)𝑛>(1+𝛿)𝜆(𝑡)𝑛𝐹(𝑥+𝑚𝜆(𝑡)𝑛𝜇)𝑃(𝑁(𝑡)=𝑛)𝐹(𝑥+𝑚𝜆(𝑡)𝜇(1+𝛿)𝜆(𝑡))𝑛>(1+𝛿)𝜆(𝑡)𝑛𝑃(𝑁(𝑡)=𝑛)𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))𝐸𝑁(𝑡)1(𝑁(𝑡)>(1+𝛿)𝜆(𝑡))=𝑜𝜆(𝑡).𝐹(𝑥+(𝑚𝜇)𝜆(𝑡))(3.26)
Substituting (3.22), (3.24), (3.25), and (3.26) into (3.4) and letting 𝛿0, the proof of (ii) is now completed.

4. Precise Large Deviations of the Prospective-Loss Process of a Quasirenewal Model

In this section we consider precise large deviations of the prospective-loss process of a quasirenewal model, where the quasi-renewal model was first introduced by Chen et al. [11]. It is a nonstandard renewal model in which innovations, modeled as real-valued random variables, are END and identically distributed, while their interarrival times are also END, identically distributed, and independent of the innovations.

Let {𝑋𝑘,𝑘=1,2,} be a sequence of END real-valued random variables with common distribution function 𝐹(𝑥)𝒞 and finite mean 𝜇, satisfying (2.7). Let {𝑁(𝑡),𝑡0} be a quasi-renewal process defined by𝑁(𝑡)=#𝑛=1,2,𝑛𝑘=1𝑌𝑘𝑡,𝑡0,(4.1) where {𝑌𝑘,𝑘=1,2,}, independent of {𝑋𝑘,𝑘=1,2,}, form a sequence of END nonnegative random variables with common distribution 𝐺 nondegenerate at zero and finite mean 1/𝜆>0. By Theorem 4.2 of [11], as 𝑡,𝑁(𝑡)𝜆𝑡1,almostsurely.(4.2) By Chen et al. [11], for any 𝛿>0, 𝑝>0, and some 𝑏>1,𝐸𝑁𝑝(𝑡)1(𝑁(𝑡)>(1+𝛿)𝜆𝑡)=𝑛>(1+𝛿)𝜆𝑡𝑛𝑝𝑃(𝑁(𝑡)=𝑛)=𝑜(1)𝑛>(1+𝛿)𝜆𝑡𝑏𝑛𝑃(𝑁(𝑡)𝑛)=𝑜(1),(4.3) where in the last step we use (4.10) in [11]. Thus, one can easily see that {𝑁(𝑡),𝑡0} satisfies Assumption 3.1. Assume that {𝑁(𝑡),𝑡0} also satisfies Assumption 3.2. Let 𝛽>0 be the safety loading coefficient. Replacing 𝑋𝑘 with 𝑋𝑘(1+𝛽)𝜇 and setting 𝑚=0 in Theorems 3.4 and 3.5, then, for any fixed 𝛾>0, the relation𝑃𝑁(𝑡)𝑘=1𝑋𝑘(1+𝛽)𝜇>𝑥𝜆𝑡𝐹(𝑥+𝛽𝜇𝜆𝑡)(4.4) holds uniformly for 𝑥𝛾𝜆𝑡.

Acknowledgments

The authors would like to thank an anonymous referee for his/her constructive and insightful comments and suggestions that greatly improved the paper. This work was partially supported by NSFC Grant 11071076, the Talents Youth Fund of Anhui Province Universities (2011SQRL012ZD), the Project Sponsored by the Doctoral Scientific Research Foundation of Anhui University, and the 211 Project of Anhui University (2009QN020B).

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