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 ๐น belongs to the class ๐’ž iflim๐‘ฆโ†˜1liminf๐‘ฅโ†’โˆž๐น(๐‘ฅ๐‘ฆ)๐น(๐‘ฅ)=1orequivalentlylim๐‘ฆโ†—1limsup๐‘ฅโ†’โˆž๐น(๐‘ฅ๐‘ฆ)๐น(๐‘ฅ)=1.(1.1) Such a distribution function ๐น is usually said to have a consistently varying tail. The heavy-tailed subclass ๐’ž was also studied by Cline and Samorodnitsky [1] who called it โ€œintermediate regular variation.โ€ Another well-known class is called the dominated variation class (denoted by ๐’Ÿ). A distribution function ๐น supported on (โˆ’โˆž,โˆž) is in ๐’Ÿ if and only iflimsup๐‘ฅโ†’โˆž๐น(๐‘ฅ๐‘ฆ)๐น(๐‘ฅ)<โˆž(1.2) for any 0<๐‘ฆ<1 (or equivalently for some 0<๐‘ฆ<1). For more details of other heavy-tailed subclasses (e.g., โ„›,๐’ฎ,โ„’, and so on) and their relations, we refer the reader to [2] or [3].

Throughout this paper, let {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} be a sequence of real-valued random variables with common distribution function 1โˆ’๐น(๐‘ฅ)=๐น(๐‘ฅ)โˆˆ๐’ž and finite mean ๐œ‡. Let {๐‘(๐‘ก),๐‘กโ‰ฅ0} be a nonnegative integer valued counting process independent of {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} with mean function ๐œ†(๐‘ก)=๐ธ๐‘(๐‘ก), which tends to infinity as ๐‘กโ†’โˆž. In insurance and finance, {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} and {๐‘(๐‘ก),๐‘กโ‰ฅ0} 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 [0,๐‘ก], can be written as๐‘†๐‘(๐‘ก)=๐‘(๐‘ก)๎“๐‘˜=1๐‘‹๐‘˜,๐‘กโ‰ฅ0.(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 ๐‘†๐‘(๐‘ก) focuses on the study of the asymptotic relation๐‘ƒ๎€ท๐‘†๐‘(๐‘ก)๎€ธโˆ’๐œ‡๐œ†(๐‘ก)>๐‘ฅโˆผ๐œ†(๐‘ก)๐น(๐‘ฅ),(1.4) which holds uniformly for some ๐‘ฅ-region ๐’Ÿ(๐‘ก) as ๐‘กโ†’โˆž. The study of precise large deviations of random sums was initiated by Klรผppelberg and Mikosch [4], who presented several applications in insurance and finance. For some latest works, we refer the reader to [2, 3, 5โ€“11], among others.

In this paper, we are interested in the deviations of random sums ๐‘†๐‘(๐‘ก) away from ๐‘š๐œ†(๐‘ก) with any fixed real number ๐‘š. We aim at proving the following asymptotic relation:๐‘ƒ๎€ท๐‘†๐‘(๐‘ก)๎€ธโˆ’๐‘š๐œ†(๐‘ก)>๐‘ฅโˆผ๐œ†(๐‘ก)๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(1.5) and the uniformity of (1.5). That is to say, in which ๐‘ฅ-region ๐’Ÿ(๐‘ก) as ๐‘กโ†’โˆž (1.5) holds uniformly. It is interesting that (1.5) reduces to (1.4) with ๐‘š replaced by ๐œ‡. Hence, we call (1.5) the extended precise large-deviation probabilities. More interestingly, setting ๐‘š=0 and replacing ๐‘‹๐‘˜ with ๐‘‹๐‘˜โˆ’(1+๐›ฟ)๐œ‡ in (1.5), where ๐›ฝ 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 {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} is extended negatively dependent (END). The END structure was firstly introduced by Liu [12].

Definition 1.1. One calls random variables {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} END if there exists a constant ๐‘€>0 such that ๐‘ƒ๎€ท๐‘‹1โ‰ค๐‘ฅ1,โ€ฆ,๐‘‹๐‘›โ‰ค๐‘ฅ๐‘›๎€ธโ‰ค๐‘€๐‘›๎‘๐‘–=1๐‘ƒ๎€ท๐‘‹๐‘–โ‰ค๐‘ฅ๐‘–๎€ธ,๐‘ƒ๎€ท๐‘‹(1.6)1>๐‘ฅ1,โ€ฆ,๐‘‹๐‘›>๐‘ฅ๐‘›๎€ธโ‰ค๐‘€๐‘›๎‘๐‘–=1๐‘ƒ๎€ท๐‘‹๐‘–>๐‘ฅ๐‘–๎€ธ(1.7) hold for each ๐‘›=1,2,โ€ฆ, and all ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›.

Recall that {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} are called ND if both (1.6) and (1.7) hold with ๐‘€=1; they are called positively dependent (PD) if inequalities (1.6) and (1.7) hold both in the reverse direction with ๐‘€=1. According to Liuโ€™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 ๐‘€ 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 {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} is an ND sequence, Liu [6] and Chen and Zhang [7] investigated precise large deviations of random sums ๐‘†๐‘(๐‘ก) 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 ๐‘†๐‘(๐‘ก) of nonnegative random variables and random sums ๐‘†๐‘๐‘ก,๐‘=โˆ‘๐‘(๐‘ก)๐‘˜=1(๐‘‹๐‘˜+๐‘) of real-valued random variables with mean zero centered by a constant ๐‘. 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 ๐‘†๐‘›=โˆ‘๐‘›๐‘˜=1๐‘‹๐‘˜. For two positive infinitesimals ๐‘“(โ‹…) and ๐‘”(โ‹…) satisfying๐‘Žโ‰คliminf๐‘“(โ‹…)๐‘”(โ‹…)โ‰คlimsup๐‘“(โ‹…)๐‘”(โ‹…)โ‰ค๐‘,(2.1) we write ๐‘“(โ‹…)=๐‘‚(๐‘”(โ‹…)) if ๐‘<โˆž; ๐‘“(โ‹…)=๐‘œ(๐‘”(โ‹…)) if ๐‘=0; ๐‘“(โ‹…)โ‰ฒ๐‘”(โ‹…) if ๐‘=1; ๐‘“(โ‹…)โ‰ณ๐‘”(โ‹…) if ๐‘Ž=1; ๐‘“(โ‹…)โˆผ๐‘”(โ‹…) if both and write ๐‘“(โ‹…)โ‰๐‘”(โ‹…) if 0<liminf(๐‘“(โ‹…)/๐‘”(โ‹…))โ‰คlimsup(๐‘“(โ‹…)/๐‘”(โ‹…))<โˆž. For theoretical and practical reasons, we usually equip them with certain uniformity. For instance, for two positive bivariate functions ๐‘“(๐‘ก,๐‘ฅ) and ๐‘”(๐‘ก,๐‘ฅ), we say that ๐‘“(๐‘ก,๐‘ฅ)โˆผ๐‘”(๐‘ก,๐‘ฅ) holds as ๐‘กโ†’โˆž uniformly for all ๐‘ฅโˆˆ๐’Ÿ(๐‘ก)โ‰ ๐œ™ in the sense thatlim๐‘กโ†’โˆžsup๐‘ฅโˆˆ๐’Ÿ(๐‘ก)||||๐‘“(๐‘ก,๐‘ฅ)||||๐‘”(๐‘ก,๐‘ฅ)โˆ’1=0.(2.2) For a distribution, set๐ฝ+๐น๎ƒฏโˆ’โˆถ=inflog๐นโˆ—(๐‘ฆ)๎ƒฐlog๐‘ฆ,๐‘ฆ>1,(2.3) where ๐นโˆ—(๐‘ฆ)=liminf๐‘ฅโ†’โˆž(๐น(๐‘ฅ๐‘ฆ))/๐น(๐‘ฅ)). In the terminology of Tang and Tsitsiashvili [13], ๐ฝ+๐น is called the upper Matuszewska index of ๐น. Clearly, if ๐นโˆˆ๐’ž, then ๐ฝ+๐น<โˆž. It holds for every ๐‘>๐ฝ+๐น that๐‘ฅโˆ’๐‘๎‚€=๐‘œ๎‚๐น(๐‘ฅ),๐‘ฅโŸถโˆž.(2.4) Moreover, ๐ฝ+๐นโ‰ฅ1 if the distribution ๐น(๐‘ฅ)=๐น(๐‘ฅ)1{๐‘ฅโ‰ฅ0} 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 {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} be a sequence of real-valued END random variables with common distribution function ๐น. If 0<๐œ‡+=๐ธ๐‘‹11{๐‘‹1โ‰ฅ0}<โˆž, then, for every fixed ๐œˆ>0 and some ๐ถ=๐ถ(๐œˆ)>0, the inequality ๐‘ƒ๎€ท๐‘†๐‘›๎€ธ>๐‘ฅโ‰ค๐‘›๐น๎‚€๐‘ฅ๐œˆ๎‚๎‚€๐‘›+๐ถ๐‘ฅ๎‚๐œˆ(2.5) holds for all ๐‘›=1,2,โ€ฆ, and ๐‘ฅ>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 {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} be a sequence of real-valued END random variables with common distribution function ๐น(๐‘ฅ)โˆˆ๐’ž and finite mean ๐œ‡, satisfying ๎‚€๐น(โˆ’๐‘ฅ)=๐‘œ๎‚||๐‘‹๐น(๐‘ฅ)as๐‘ฅโŸถโˆž,๐ธ1||๐‘Ÿ1{๐‘‹1โ‰ค0}<โˆžforsome๐‘Ÿ>1.(2.6) Then, for any fixed ๐›พ>0, relation ๐‘ƒ๎€ท๐‘†๐‘›๎€ธโˆ’๐‘›๐œ‡>๐‘ฅโˆผ๐‘›๐น(๐‘ฅ),as๐‘›โŸถโˆž,(2.7) holds uniformly for all ๐‘ฅโ‰ฅ๐›พ๐‘›.

3. Main Results and Their Proofs

In this sequel, all limiting relationships, unless otherwise stated, are according to ๐‘กโ†’โˆž. To state the main results, we need the following two basic assumptions on the counting process {๐‘(๐‘ก),๐‘กโ‰ฅ0}.

Assumption 3.1. For any ๐›ฟ>0 and some ๐‘>๐ฝ+๐น, ๐ธ๐‘๐‘(๐‘ก)1(๐‘(๐‘ก)>(1+๐›ฟ)๐œ†(๐‘ก))=๐‘‚(๐œ†(๐‘ก)).(3.1)

Assumption 3.2. The relation ๎‚€๐‘ƒ(๐‘(๐‘ก)โ‰ค(1โˆ’๐›ฟ)๐œ†(๐‘ก))=๐‘œ๐œ†(๐‘ก)๎‚๐น(๐œ†(๐‘ก))(3.2) holds for all 0<๐›ฟ<1.

Remark 3.3. One can easily see that Assumption 3.1 or Assumption 3.2 implies that ๐‘(๐‘ก)๐œ†(๐‘ก)๐‘ƒโŸถ1.(3.3) See [5, 11].

Theorem 3.4. Let {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} be a sequence of END real-valued random variables with common distribution function ๐น(๐‘ฅ)โˆˆ๐’ž having finite mean ๐œ‡โ‰ฅ0 and satisfying (2.6), and let {๐‘(๐‘ก),๐‘กโ‰ฅ0} be a nonnegative integer-valued counting process independent of {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} satisfying Assumption 3.1. Let ๐‘š be a real number; then, for any ๐›พ>(๐œ‡โˆ’๐‘š)โˆจ0, the relation (1.5) holds uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก).

Theorem 3.5. Let {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} be a sequence of END real-valued random variables with common distribution function ๐น(๐‘ฅ)โˆˆ๐’ž having finite mean ๐œ‡<0 and satisfying (2.6), and let {๐‘(๐‘ก),๐‘กโ‰ฅ0} be a nonnegative integer valued counting process independent of {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ}. (i)Assume that {๐‘(๐‘ก),๐‘กโ‰ฅ0} satisfies Assumption 3.1 and ๐‘š is a real number (regardless of ๐‘šโ‰ฅ0 or ๐‘š<0), then for any fixed ๐›พ>โˆ’๐‘šโˆจ0, the relation (1.5) holds uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก).(ii)Assume that {๐‘(๐‘ก),๐‘กโ‰ฅ0} satisfies Assumption 3.2 and ๐‘š is a negative real number; then, for any fixed ๐›พโˆˆ(๐œ‡โˆ’๐‘šโˆจ0,โˆ’๐‘š], the relation (1.5) holds uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก).

Remark 3.6. One can easily see that Theorem 3.4 extends Theorem 3.1 of [11] with ๐‘š replaced by ๐œ‡. On the other hand, replacing ๐‘‹๐‘˜ with ๐‘‹๐‘˜โˆ’๐œ‡+๐‘, setting ๐‘š=0, and noticing that ๐ธ(๐‘‹๐‘˜โˆ’๐œ‡+๐‘)=๐‘, (3.4) yields Theorem 4.1(i) of [11].

Remark 3.7. Under the conditions of Theorem 3.5, choosing ๐‘š=๐œ‡, one can easily see that the relation (1.4) holds uniformly over the ๐‘ฅ-region ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก) for arbitrarily fixed ๐›พ>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: ๐‘ƒ๎€ท๐‘†๐‘(๐‘ก)๎€ธ=โˆ’๐‘š๐œ†(๐‘ก)>๐‘ฅโˆž๎“๐‘›=1๐‘ƒ๎€ท๐‘†๐‘›๎€ธ=๎ƒฉ๎“โˆ’๐‘š๐œ†(๐‘ก)>๐‘ฅ๐‘ƒ(๐‘(๐‘ก)=๐‘›)๐‘›<(1โˆ’๐›ฟ)๐œ†(๐‘ก)+๎“(1โˆ’๐›ฟ)๐œ†(๐‘ก)โ‰ค๐‘›โ‰ค(1+๐›ฟ)๐œ†(๐‘ก)+๎“๐‘›>(1+๐›ฟ)๐œ†(๐‘ก)๎ƒช๎€ท๐‘†ร—๐‘ƒ๐‘›๎€ธโˆ’๐‘š๐œ†(๐‘ก)>๐‘ฅ๐‘ƒ(๐‘(๐‘ก)=๐‘›)โˆถ=๐ผ1(๐‘ฅ,๐‘ก)+๐ผ2(๐‘ฅ,๐‘ก)+๐ผ3(๐‘ฅ,๐‘ก),(3.4) where we choose 0<๐›ฟ<1 such that (๐›พ+๐‘š)/(1+๐›ฟ)โˆ’๐œ‡>0.
We first deal with ๐ผ1(๐‘ฅ,๐‘ก). Note that ๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡โ‰ฅ((๐›พ+๐‘š)/(1โˆ’๐›ฟ)โˆ’๐œ‡)๐‘›. Thus, as ๐‘กโ†’โˆž and uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก), it follows from Lemma 2.2 that๐ผ1(๎“๐‘ฅ,๐‘ก)โˆผ๐‘›<(1โˆ’๐›ฟ)๐œ†(๐‘ก)๐‘›โ‰ค๎“๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡)๐‘ƒ(๐‘(๐‘ก)=๐‘›)๐‘›<(1โˆ’๐›ฟ)๐œ†(๐‘ก)โ‰ค๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’(1โˆ’๐›ฟ)๐œ‡๐œ†(๐‘ก))๐‘›๐‘ƒ(๐‘(๐‘ก)=๐‘›)(1โˆ’๐›ฟ)๐œ†(๐‘ก)๐น๎‚€(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))๐‘ƒ(๐‘(๐‘ก)<(1โˆ’๐›ฟ)๐œ†(๐‘ก))=๐‘œ๐œ†(๐‘ก)๎‚.๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(3.5)
Next, for ๐ผ2(๐‘ฅ,๐‘ก), noticing that ๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡โ‰ฅ((๐›พ+๐‘š)/(1+๐›ฟ)โˆ’๐œ‡)๐‘›, as ๐‘กโ†’โˆž and uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก), Lemma 2.2 yields that๐ผ2(๎“๐‘ฅ,๐‘ก)โˆผ(1โˆ’๐›ฟ)๐œ†(๐‘ก)โ‰ค๐‘›โ‰ค(1+๐›ฟ)๐œ†(๐‘ก)๐‘›๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡)๐‘ƒ(๐‘(๐‘ก)=๐‘›)โ‰ค(1+๐›ฟ)๐œ†(๐‘ก)๎ƒฉ||||๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐œ‡(1+๐›ฟ)๐œ†(๐‘ก))๐‘ƒ๐‘(๐‘ก)||||๎ƒช๐œ†(๐‘ก)โˆ’1<๐›ฟโˆผ(1+๐›ฟ)๐œ†(๐‘ก)๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก)โˆ’๐›ฟ๐œ‡๐œ†(๐‘ก))โ‰ค(1+๐›ฟ)๐œ†(๐‘ก)๐น๎‚ต๎‚ต1โˆ’๐›ฟ๐œ‡๎‚ถ๎‚ถ.๐›พ+๐‘šโˆ’๐œ‡(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(3.6) On the other hand, ๐ผ2(๎“๐‘ฅ,๐‘ก)โˆผ(1โˆ’๐›ฟ)๐œ†(๐‘ก)โ‰ค๐‘›โ‰ค(1+๐›ฟ)๐œ†(๐‘ก)๐‘›๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡)๐‘ƒ(๐‘(๐‘ก)=๐‘›)โ‰ค(1โˆ’๐›ฟ)๐œ†(๐‘ก)๎ƒฉ||||๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐œ‡(1โˆ’๐›ฟ)๐œ†(๐‘ก))๐‘ƒ๐‘(๐‘ก)||||๎ƒช๐œ†(๐‘ก)โˆ’1<๐›ฟโˆผ(1โˆ’๐›ฟ)๐œ†(๐‘ก)๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก)+๐›ฟ๐œ‡๐œ†(๐‘ก))โ‰ค(1โˆ’๐›ฟ)๐œ†(๐‘ก)๐น๎‚ต๎‚ต1+๐›ฟ๐œ‡๎‚ถ๎‚ถ.๐›พ+๐‘šโˆ’๐œ‡(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(3.7)
Finally, to deal with ๐ผ3(๐‘ฅ,๐‘ก), we formulate the remaining proof into two parts according to ๐‘šโ‰ฅ0 and ๐‘š<0. In the case of ๐‘šโ‰ฅ0, setting ๐œˆ=๐‘ in Lemma 2.1 with ๐‘>๐ฝ+๐นโ‰ฅ1, for sufficiently large ๐‘ก and ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก), there exists some constant ๐ถ1>0 such that๐ผ3(๎“๐‘ฅ,๐‘ก)โ‰ค๐‘›>(1+๐›ฟ)๐œ†(๐‘ก)๐‘ƒ๎€ท๐‘†๐‘›๎€ธโ‰ค๎“>๐‘ฅ๐‘ƒ(๐‘(๐‘ก)=๐‘›)๐‘›>(1+๐›ฟ)๐œ†(๐‘ก)๎‚ต๐‘›๐น๎‚ต๐‘ฅ๐‘๎‚ถ+๐ถ1๎‚€๐‘›๐‘ฅ๎‚๐‘๎‚ถ๐‘ƒ(๐‘(๐‘ก)=๐‘›).(3.8) In the case of ๐‘š<0, note that 1+๐‘š/๐›พ>0 since ๐›พ>๐œ‡โˆ’๐‘šโ‰ฅโˆ’๐‘š>0. Similar to (3.8), for sufficiently large ๐‘ก and ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก), there exists some constant ๐ถ2>0 such that ๐ผ3(๎“๐‘ฅ,๐‘ก)โ‰ค๐‘›>(1+๐›ฟ)๐œ†(๐‘ก)๐‘ƒ๎€ท๐‘†๐‘›๎€ธโ‰ค๎“>(1+๐‘š/๐›พ)๐‘ฅ๐‘ƒ(๐‘(๐‘ก)=๐‘›)๐‘›>(1+๐›ฟ)๐œ†(๐‘ก)๎‚ต๐‘›๐น๎‚ต(1+๐‘š/๐›พ)๐‘ฅ๐‘๎‚ถ+๐ถ2๎‚ต๐‘›๎‚ถ(1+๐‘š/๐›พ)๐‘ฅ๐‘๎‚ถ๐‘ƒ(๐‘(๐‘ก)=๐‘›).(3.9) As a result, by (2.4), as ๐‘กโ†’โˆž and uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก), both (3.8) and (3.9) yield that ๐ผ3(๐‘ฅ,๐‘ก)โ‰ฒ๐น(๐‘ฅ)๐ธ๐‘(๐‘ก)1{๐‘(๐‘ก)>(1+๐›ฟ)๐œ†(๐‘ก)}+๐‘ฅโˆ’๐‘๐ธ๐‘๐‘(๐‘ก)1{๐‘(๐‘ก)>(1+๐›ฟ)๐œ†(๐‘ก)}โ‰ค๐‘œ(1)๐น(๐‘ฅ)๐ธ๐‘๐‘(๐‘ก)1{๐‘(๐‘ก)>(1+๐›ฟ)๐œ†(๐‘ก)}โ‰ค๐‘œ(1)๐œ†(๐‘ก)๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก)),(3.10) where in the last step, we used ๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))โ‰ฅ๐น||||๎‚ต๎‚ต1+๐‘šโˆ’๐œ‡๐›พ๎‚ถ๐‘ฅ๎‚ถโ‰ณ๐น(๐‘ฅ).(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 ๐นโˆˆ๐’ž and the arbitrariness of ๐›ฟ.

Proof of Theorem 3.5. (i) We also start with the decomposition (3.4).
For ๐ผ1(๐‘ฅ,๐‘ก) and ๐ผ2(๐‘ฅ,๐‘ก), note that ๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡โ‰ฅ((๐›พ+๐‘š)/(1+๐›ฟ)โˆ’๐œ‡)๐‘› since ๐‘›โ‰ค(1+๐›ฟ)๐œ†(๐‘ก). Hence, mimicking the proof of Theorem 3.4, we obtain, as ๐‘กโ†’โˆž and uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก),๐ผ1(๎“๐‘ฅ,๐‘ก)โˆผ๐‘›<(1โˆ’๐›ฟ)๐œ†(๐‘ก)๐‘›โ‰ค๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡)๐‘ƒ(๐‘(๐‘ก)=๐‘›)๎“๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก))๐‘›<(1โˆ’๐›ฟ)๐œ†(๐‘ก)๎‚€๐‘›๐‘ƒ(๐‘(๐‘ก)=๐‘›)=๐‘œ๐œ†(๐‘ก)๎‚,๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(3.12) where, in the last step, we used the relation ๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก))โ‰ค๐น๐‘š๎‚ต๎‚ต1+๐›พ๎‚ถ๐‘ฅ๎‚ถโ‰ฒ๐น(๐‘ฅ),(3.13)๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))โ‰ฅ๐น||||๎‚ต๎‚ต1+๐‘šโˆ’๐œ‡๐›พ๎‚ถ๐‘ฅ๎‚ถโ‰ณ๐น(๐‘ฅ).(3.14)
Again, as ๐‘กโ†’โˆž and uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก),๐ผ2(๎“๐‘ฅ,๐‘ก)โˆผ(1โˆ’๐›ฟ)๐œ†(๐‘ก)โ‰ค๐‘›โ‰ค(1+๐›ฟ)๐œ†(๐‘ก)๐‘›๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡)๐‘ƒ(๐‘(๐‘ก)=๐‘›)โ‰ค(1+๐›ฟ)๐œ†(๐‘ก)๎ƒฉ||||๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐œ‡(1โˆ’๐›ฟ)๐œ†(๐‘ก))๐‘ƒ๐‘(๐‘ก)||||๎ƒช๐œ†(๐‘ก)โˆ’1<๐›ฟโ‰ฒ(1+๐›ฟ)๐œ†(๐‘ก)๐น๎‚ต๎‚ต1+๐›ฟ๐œ‡๎‚ถ(๎‚ถ,๐ผ๐›พ+๐‘šโˆ’๐œ‡๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))2(๐‘ฅ,๐‘ก)โ‰ณ(1โˆ’๐›ฟ)๐œ†(๐‘ก)๐น๎‚ต๎‚ต1โˆ’๐›ฟ๐œ‡๎‚ถ๎‚ถ.๐›พ+๐‘šโˆ’๐œ‡(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(3.15)
Finally, in ๐ผ3(๐‘ฅ,๐‘ก), setting ๐œˆ=๐‘ in Lemma 2.1 with ๐‘>๐ฝ+๐นโ‰ฅ1, by (2.4) and Assumption 3.1, as ๐‘กโ†’โˆž and uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก), there exists a constant ๐ถ2>0 such that๐ผ3(๎“๐‘ฅ,๐‘ก)โ‰ค๐‘›>(1+๐›ฟ)๐œ†(๐‘ก)๐‘ƒ๎€ท๐‘†๐‘›๎€ธโ‰ค๎“>(1+๐‘š/๐›พ)๐‘ฅ๐‘ƒ(๐‘(๐‘ก)=๐‘›)๐‘›>(1+๐›ฟ)๐œ†(๐‘ก)๎‚ต๐‘›๐น๎‚ต(1+๐‘š/๐›พ)๐‘ฅ๐‘๎‚ถ+๐ถ2๎‚ต๐‘›๎‚ถ(1+๐‘š/๐›พ)๐‘ฅ๐‘๎‚ถโ‰๐‘ƒ(๐‘(๐‘ก)=๐‘›)๐น(๐‘ฅ)๐ธ๐‘(๐‘ก)1{๐‘(๐‘ก)>(1+๐›ฟ)๐œ†(๐‘ก)}+๐‘ฅโˆ’๐‘๐ธ๐‘๐‘(๐‘ก)1{๐‘(๐‘ก)>(1+๐›ฟ)๐œ†(๐‘ก)}โ‰ค๐‘œ(1)๐œ†(๐‘ก)๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก)),(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 ๐นโˆˆ๐’ž and the arbitrariness of ๐›ฟ.
(ii) We also start with the representation (3.4) in which we choose 0<๐›ฟ<1 such that (๐›พ+๐‘š)/(1โˆ’๐›ฟ)โˆ’๐œ‡>0.
To deal with ๐ผ1(๐‘ฅ,๐‘ก), arbitrarily choosing ๐›พ1>โˆ’๐‘š, we split the ๐‘ฅ-region into two disjoint regions as[๎€บ๐›พ๐›พ๐œ†(๐‘ก),โˆž)=1๐œ†๎€ธโˆช๎€บ(๐‘ก),โˆž๐›พ๐œ†(๐‘ก),๐›พ1๐œ†๎€ธ(๐‘ก).(3.17) For the first ๐‘ฅ-region ๐‘ฅโ‰ฅ๐›พ1๐œ†(๐‘ก), noticing that ๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡>|๐œ‡|๐‘›, by Lemma 2.2, it holds uniformly for all ๐‘ฅโ‰ฅ๐›พ1๐œ†(๐‘ก) that ๐ผ1(๎“๐‘ฅ,๐‘ก)โˆผ๐‘›<(1โˆ’๐›ฟ)๐œ†(๐‘ก)๐‘›๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡)๐‘ƒ(๐‘(๐‘ก)=๐‘›)โ‰ค(1โˆ’๐›ฟ)๐œ†(๐‘ก)๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก))๐‘ƒ(๐‘(๐‘ก)โ‰ค(1โˆ’๐›ฟ)๐œ†(๐‘ก))โ‰ค(1โˆ’๐›ฟ)๐œ†(๐‘ก)๐น(๐‘ฅ)๐‘ƒ(๐‘(๐‘ก)โ‰ค(1โˆ’๐›ฟ)๐œ†(๐‘ก))โ‰ค(1โˆ’๐›ฟ)๐œ†(๐‘ก)๎‚€๐œ†๐น(๐‘ฅ(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))๐‘ƒ(๐‘(๐‘ก)โ‰ค(1โˆ’๐›ฟ)๐œ†(๐‘ก))=๐‘œ(๐‘ก)๐น๎‚,(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(3.18) where the second step and the before the last before the step can be verified, respectively, as ๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก))โ‰ค๐น๐‘š๎‚ต๎‚ต1+๐›พ1๎‚ถ๐‘ฅ๎‚ถโ‰ฒ๐น(๐‘ฅ),๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))โ‰ฅ๐น||||๎‚ต๎‚ต1+๐‘šโˆ’๐œ‡๐›พ1๎‚ถ๐‘ฅ๎‚ถโ‰ณ๐น(๐‘ฅ).(3.19) For the second ๐‘ฅ-region ๐›พ๐œ†(๐‘ก)โ‰ค๐‘ฅ<๐›พ1๐œ†(๐‘ก), note that ๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))โ‰ฅ๐น||||๎‚ต๎‚ต1+๐‘šโˆ’๐œ‡๐›พ๎‚ถ๐‘ฅ๎‚ถโ‰๐น(๐‘ฅ)โ‰ฅ๐น๎€ท๐›พ1๎€ธโ‰๐œ†(๐‘ก)๐น(๐œ†(๐‘ก)).(3.20) Hence, by Assumption 3.2, we still obtain ๐ผ1(๐‘ฅ,๐‘ก)โ‰ค๐‘ƒ(๐‘(๐‘ก)โ‰ค(1โˆ’๐›ฟ)๐œ†(๐‘ก))=๐‘œ(1)๐œ†(๐‘ก)๎‚€๐น(๐œ†(๐‘ก))โ‰ฒ๐‘œ๐œ†(๐‘ก)๎‚๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(3.21) uniformly for all ๐›พ๐œ†(๐‘ก)โ‰ค๐‘ฅ<๐›พ1๐œ†(๐‘ก). As a result, the relation ๐ผ1๎‚€(๐‘ฅ,๐‘ก)=๐‘œ๐œ†(๐‘ก)๎‚๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(3.22) holds uniformly for all ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก).
For ๐ผ2(๐‘ฅ,๐‘ก), since ๐›พโˆˆ(๐œ‡โˆ’๐‘šโˆจ0,โˆ’๐‘š], it holds that๎‚€๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡โ‰ฅ(๐›พ+๐‘š)๐œ†(๐‘ก)โˆ’๐œ‡๐‘›โ‰ฅ๐›พ+๐‘š๎‚1โˆ’๐›ฟโˆ’๐œ‡๐‘›.(3.23) It follows from Lemma 2.2 that for all ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก)๐ผ2(๎“๐‘ฅ,๐‘ก)โˆผ(1โˆ’๐›ฟ)๐œ†(๐‘ก)โ‰ค๐‘›โ‰ค(1+๐›ฟ)๐œ†(๐‘ก)๐‘›๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡)๐‘ƒ(๐‘(๐‘ก)=๐‘›)โ‰ค(1+๐›ฟ)๐œ†(๐‘ก)๎ƒฉ||||๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐œ‡(1โˆ’๐›ฟ)๐œ†(๐‘ก))๐‘ƒ๐‘(๐‘ก)||||๎ƒช๐œ†(๐‘ก)โˆ’1<๐›ฟโˆผ(1+๐›ฟ)๐œ†(๐‘ก)๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก)+๐›ฟ๐œ‡๐œ†(๐‘ก))โ‰ค(1+๐›ฟ)๐œ†(๐‘ก)๐น๎‚ต๎‚ต1+๐›ฟ๐œ‡๎‚ถ๎‚ถ.๐›พ+๐‘šโˆ’๐œ‡(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(3.24) Symmetrically, ๐ผ2(๐‘ฅ,๐‘ก)โ‰ณ(1โˆ’๐›ฟ)๐œ†(๐‘ก)๐น๎‚ต๎‚ต1โˆ’๐›ฟ๐œ‡๎‚ถ๎‚ถ๐›พ+๐‘šโˆ’๐œ‡(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก)).(3.25)
Finally, in ๐ผ3(๐‘ฅ,๐‘ก), note that ๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡โ‰ฅ(๐›พ+๐‘š)๐œ†(๐‘ก)โˆ’๐œ‡๐‘›โ‰ฅ((๐›พ+๐‘š)/(1+๐›ฟ)โˆ’๐œ‡)๐‘›. Therefore, Lemma 2.2 implies, as ๐‘กโ†’โˆž and uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†(๐‘ก), that๐ผ3(๎“๐‘ฅ,๐‘ก)โˆผ๐‘›>(1+๐›ฟ)๐œ†(๐‘ก)๐‘›โ‰ค๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐‘›๐œ‡)๐‘ƒ(๐‘(๐‘ก)=๐‘›)๎“๐น(๐‘ฅ+๐‘š๐œ†(๐‘ก)โˆ’๐œ‡(1+๐›ฟ)๐œ†(๐‘ก))๐‘›>(1+๐›ฟ)๐œ†(๐‘ก)โ‰ค๐‘›๐‘ƒ(๐‘(๐‘ก)=๐‘›)๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))๐ธ๐‘(๐‘ก)1(๐‘(๐‘ก)>(1+๐›ฟ)๐œ†(๐‘ก))๎‚€=๐‘œ๐œ†(๐‘ก)๎‚.๐น(๐‘ฅ+(๐‘šโˆ’๐œ‡)๐œ†(๐‘ก))(3.26)
Substituting (3.22), (3.24), (3.25), and (3.26) into (3.4) and letting ๐›ฟโ†“0, 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 {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ} be a sequence of END real-valued random variables with common distribution function ๐น(๐‘ฅ)โˆˆ๐’ž and finite mean ๐œ‡, satisfying (2.7). Let {๐‘(๐‘ก),๐‘กโ‰ฅ0} be a quasi-renewal process defined by๎ƒฏ๐‘(๐‘ก)=#๐‘›=1,2,โ€ฆโˆถ๐‘›๎“๐‘˜=1๐‘Œ๐‘˜๎ƒฐโ‰ค๐‘ก,๐‘กโ‰ฅ0,(4.1) where {๐‘Œ๐‘˜,๐‘˜=1,2,โ€ฆ}, independent of {๐‘‹๐‘˜,๐‘˜=1,2,โ€ฆ}, form a sequence of END nonnegative random variables with common distribution ๐บ nondegenerate at zero and finite mean 1/๐œ†>0. By Theorem 4.2 of [11], as ๐‘กโ†’โˆž,๐‘(๐‘ก)๐œ†๐‘กโŸถ1,almostsurely.(4.2) By Chen et al. [11], for any ๐›ฟ>0, ๐‘>0, and some ๐‘>1,๐ธ๐‘๐‘(๐‘ก)1(๐‘(๐‘ก)>(1+๐›ฟ)๐œ†๐‘ก)=๎“๐‘›>(1+๐›ฟ)๐œ†๐‘ก๐‘›๐‘๎“๐‘ƒ(๐‘(๐‘ก)=๐‘›)=๐‘œ(1)๐‘›>(1+๐›ฟ)๐œ†๐‘ก๐‘๐‘›๐‘ƒ(๐‘(๐‘ก)โ‰ฅ๐‘›)=๐‘œ(1),(4.3) where in the last step we use (4.10) in [11]. Thus, one can easily see that {๐‘(๐‘ก),๐‘กโ‰ฅ0} satisfies Assumption 3.1. Assume that {๐‘(๐‘ก),๐‘กโ‰ฅ0} also satisfies Assumption 3.2. Let ๐›ฝ>0 be the safety loading coefficient. Replacing ๐‘‹๐‘˜ with ๐‘‹๐‘˜โˆ’(1+๐›ฝ)๐œ‡ and setting ๐‘š=0 in Theorems 3.4 and 3.5, then, for any fixed ๐›พ>0, the relation๐‘ƒ๎ƒฉ๐‘(๐‘ก)๎“๐‘˜=1๎€ท๐‘‹๐‘˜๎€ธ๎ƒชโˆ’(1+๐›ฝ)๐œ‡>๐‘ฅโˆผ๐œ†๐‘ก๐น(๐‘ฅ+๐›ฝ๐œ‡๐œ†๐‘ก)(4.4) holds uniformly for ๐‘ฅโ‰ฅ๐›พ๐œ†๐‘ก.

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).