Abstract

We consider a dependent multirisk model in insurance, where all the claims constitute a linearly extended negatively orthant dependent (LENOD) random array, and then upper and lower bounds for precise large deviations of nonrandom and random sums of random variables with dominated variation are investigated. The obtained results extend some related existing ones.

1. Introduction

In a classic insurance risk model the surplus is described as the initial surplus plus the premium income with the claims taken off. Since asymptotic behavior for precise large deviations of the loss process of insurance risk models has theoretical significance and extensive applications, it has been widely investigated and there appeared to be a great deal of research literature. Some earlier works on precise large deviations can be found in Nagaev [1, 2], Heyde [3, 4], and Nagaev [5, 6], among others. Recent works on this field can be found in Cline and Hsing [7], Klüppelberg and Mikosch [8], Tang et al. [9], Tang [10], Liu [11], Chen and Zhang [12], Shen and Lin [13], Liu [14], Chen et al. [15], and Chen and Yuen [16], among others.

In consideration of insurance reality, some researchers have begun to focus on the precise large deviations of multi-risk models in the past few years. See S. Wang and W. Wang [17], Lu [18, 19], He at al. [20], and S. J. Wang and W. S. Wang [21], among others. For convenience of representation, we adopt the notations of S. Wang and W. Wang [17]. Assume that the insurer manages types of insurance contracts at the same time, where is any fixed positive integer. The th related loss amounts (claims) are denoted by random variables with common distribution satisfying for all . Let be independent nonnegative integer-valued counting processes independent of , satisfying as . In insurance multi-risk models, always denote the th claim numbers of the related insurance contracts. Obviously, all the claims constitute the following random array: Therefore, the loss process of the insurer during the period can be described as . If the random arrays (1) are independent (i.e., to say all random variables are independent) and all the claims have consistently varying tails (see the definition below), under some mild conditions, S. Wang and W. Wang [17] obtained the following result. For any fixed , as , the relation holds uniformly for .

The relation (2) describes the so-called precise large deviations for random sums in multi-risk models. Subsequently, Lu [18, 19] investigated precise large deviations with subexponential and long-tailed claims in multi-risk models under independent structures, respectively. He et al. [20] obtained the lower bounds of precise large deviations of multi-risk models with nonnegative random variables (regardless of heavy or light tails) under a specific dependence structure. However, on the one hand, for practical reasons, the independence assumptions in the above-mentioned papers are quite unrealistic. On the other hand, up to now, most works on precise large deviations with heavy tails concentrated on the consistent variation or some other heavy-tailed subclasses although some specific dependence structures have been considered. Therefore, it is more interesting to study the estimation of precise large deviation probabilities of aggregate claims in the presence of dominated variation and some dependence structures, where dominated variation strictly includes consistent variation. More recently, some researchers have begun to focus on this issue and obtained some interesting results. See Wang et al. [22], Wang et al. [23], Yang et al. [24], Yang and Wang [25], and Chen and Qu [26], among others. Motivated by the two reasons mentioned above, in this paper, upper and lower bounds for precise large deviations of aggregate claims with dominated variation in dependent multi-risk models (see the dependent structures in Definitions 1 and 2 below) are investigated. The obtained results extend some related existing ones.

To close this section, we introduce two new dependence structures called ENOD and LENOD, respectively, which are the basic assumptions in this paper. The idea of Definition 1 comes from Liu [14].

Definition 1. We call random arrays (1)Extended Negatively Lower Orthant Dependent (ENLOD) if there exists some constant such that for each , , , and all , (2)Extended Negatively Upper Orthant Dependent (ENUOD) if there exists some constant such that for each , , , and all , (3)Extended Negatively Orthant Dependent (ENOD) if both (3) and (4) hold for each , , , and all .

Definition 2. We call random arrays (1)Linearly Extended Negatively Lower Orthant Dependent (LENLOD) if for each , any finite disjoint subsets of , and positive ’s, (2)Linearly Extended Negatively Upper Orthant Dependent (LENUOD) if for each , any finite disjoint subsets of , and positive ’s, (3)Linearly Extended Negatively Orthant Dependent (LENOD) if both (5) and (6) hold for each , any finite disjoint subsets of and positive ’s.

Remark 3. The idea of Definition 2 is due to Newman [27] who first put forward the concept of LNQD (linearly negative quadrant dependent) when studying the central limit theorem. Other related dependence structures are called NQD and NA. See Joag-Dev and Proschan [28] for more details. It is well known that NA implies LNQD and LNQD is slightly stronger than NQD. In the consideration of the need of studying multivariate random variables, we introduce the concept of LENOD random arrays in this paper. By definitions, one can also easily check that NA random arrays must be LENOD ones and LENOD implies ENQD. Moreover, LENOD is more comprehensive than NA in that it can reflect not only a negative association structure but also a positive one to some extent.

Remark 4. It is worth to mention that there maybe exist some flaws in the definition of extended negatively associated (ENA) structure of S. J. Wang and W. S. Wang [21]. In fact, S. J. Wang and W. S. Wang [21] stated that the relationship holds for fixed and any pair of coordinatewise increasing functions . Considering the case that , inequality (7) can not hold because It might be questionable. Therefore, in this paper, we redefine a new dependence structure called LENOD for random arrays to avoid this problem. Fortunately, under this new dependence structure, the main results still hold and extend some related existing ones.

The rest of this paper is organized as follows. Section 2 gives some preliminaries. Precise large deviations for nonrandom sums and random sums with dominated variation in dependent multi-risk models are presented in Sections 3 and 4.

2. Preliminaries

For convenience, hereafter, for two positive infinitesimals and satisfying we write if ; if . For two positive bivariate functions and , we say holds as uniformly for all in the sense that .

For simplicity, for any positive integer sequence , we use the following notations:

2.1. Heavy-Tailed Distributions

In risk theory, heavy-tailed distribution functions 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 nonnegative random variable (or its distribution function ) is heavy-tailed if it has no finite exponential moments. For details, we refer the reader to Embrechts, Klüppelberg, and Mikosch [8], among others. For convenience of use, we recall some important subclasses of heavy-tailed distributions. A quite large subclass is called long-tailed distribution class denoted by . A distribution function is said to belong to if for all Another important subclass is called the class of random variables with dominatedly varying tails (or the class of distribution functions with dominated variation) denoted by . We say that a distribution function is in if for any (or equivalently for some ) Furthermore, for any , set and then define In the terminology of Tang and Tsitsiashvili [29],   is called the (upper) Matuszewska index of . The following proposition is well known.

Proposition 5. Consider   for all   for some .

Some other subclasses are as follows. Denote by the class of random variables with consistently varying tails (or distribution functions with consistent variation). For some , denote by for all the extended regularly varying class. Particularly, if , it reduces to the regularly varying class, denoted by . For the heavy-tailed distribution subclasses mentioned above, it is well known that the following inclusions hold:

Finally, we define an important quantity of heavy-tailed distribution function needed in the main results.

Definition 6. For any heavy-tailed distribution and , define

Remark 7. Obviously, . Furthermore, if , then for any by the definition of .

2.2. Some Lemmas

In this sequel, we will give some lemmas needed in the proofs of the main results. Lemmas 1 and 2 are due to S. J. Wang and W. S. Wang [21] (Lemmas 2.3 and 2.5). To state the results, we should introduce two assumptions added on the process , which also appeared in Chen et al. [15] and S. J. Wang and W. S. Wang [21].

Assumption A. For any and some , as ,

Assumption B. For all , as ,

Lemma 1. Let be ENOD random variables with common distribution and finite mean , satisfying for some and then, for any , as , the relation holds uniformly for .

Lemma 2. Let be a sequence of ENOD random variables with common distribution function and finite mean , satisfying (18), and let be a nonnegative integer-valued counting process independent of , satisfying Assumptions A and B; then, for any , as , the relation holds uniformly for .

Lemmas 3–5 play important roles in the proofs of main results and have their own interests. Lemma 3 describes an important property of the quantity defined in Definition 6. Lemmas 4 and 5 present precise large deviations for nonrandom and random sums of random variables with dominated variation in single-risk models, respectively.

Lemma 3. Let any distribution , and, for any fixed real number , denote the distribution by . Then,(i)  when  ;(ii)  when  .

Proof. (i) By the definition of , one can easily check that .
When , we get Conversely, for any and large enough , noticing that , we have
(ii) For any and fixed real number , by Definition 6, one can easily obtain . Thus, when , (21) and (22) imply and . It ends the proof of Lemma 3.

Remark 8. It is interesting that Lemma 3 indicates that the quantity maybe will not be equal to when we let for any fixed real number . In essence, it is due to the fact that is not a subset of . Thus, we should be very cautious when dealing with the random variables from the subclass .

Lemma 4. Let be a sequence of ENOD random variables with common distribution and finite mean , satisfying Then, for any fixed , where .

Proof. If , one can mimic the proof of  Theorem 1.1 of Liu [14] to prove (24). Also one can get the relation (24) directly from Corollaries 1 and 2 of Wang et al. [23]. Therefore, we omit the process here.
For the case of , let ,  and note that is distributed by (see the notation in Lemma 3). Therefore, if , by the fact that and Lemma 3, it yields that Combining (25) and (26), we get (24). Similarly as above, one can also check that the relation (24) holds if and this ends the proof of Lemma 4.

Lemma 5. Let be a sequence of ENOD random variables with common distribution and finite mean , satisfying (23), and let be a nonnegative integer-valued counting process independent of . Then, for any fixed , the relation holds under one of the following two conditions (where is defined in Lemma 4):(i)when , Assumption A holds;(ii)when , Assumption B holds.

To prove Lemma 5, one can mimic the proof of Theorem 3.1 of S. Wang and X. Wang, [30] by replacing by and using Lemma 3. For simplicity, we also omit it here.

3. Large Deviations for Nonrandom Sums

In this section, we will give precise large deviations for nonrandom sums with dominated variation in multi-risk models under LENOD structures. Hereafter, always represents a finite and positive constant whose value may vary in different places.

Theorem 9. Let be LENOD random arrays. For all , have common distribution function and finite expectation , satisfying If for some and for all , then, for any fixed , we have, for the lower bound, and, for the upper bound, where and .

Remark 10. In Theorem 9, if we assume , then (29) and (30) hold with for all . Particularly, if for all , then . Hence, Theorem 9 reduces to Theorem  3.1 of S. J. Wang and W. S. Wang [21]. Moreover, if we also assume are nonnegative independent random arrays, one can easily see the condition (28) naturally holds. Therefore, (29) and (30) constitute the results of Theorem  3.1 of S. Wang and W. Wang [17].

Remark 11. If we suppose or all are the same distribution functions, then Theorem 9 implies Lemma 4. It means that Theorem 9 extends the result of Lemma 4 to multi-risk models. Particularly, if we also assume , then Theorem 9 reduces to Theorem 2.1 of Liu [14].

Proof. As usual, we use induction to prove Theorem 9. For the case of , we first show that By using the same decomposition method of S. J. Wang and W. S. Wang [21], for any and any , it holds that For and , by the same argument of relation (3.7) of S. J. Wang and W. S. Wang [21], and Lemmas 1 and 4, one can easily obtain, for any sufficiently small , that there exist sufficiently large such that, uniformly for , Next we turn to estimate . Noticing that random arrays are LENOD, again by Lemma 4, for sufficiently large and uniformly for , we arrive at where in the last step we use the fact that . Furthermore, note that for ; hence, for sufficiently small and sufficiently large , Combining (32)–(36), it holds that the left-hand side of (32) is bounded from below by Therefore, (31) can be derived directly from above by the arbitrariness of .
Next we show that For any and , Lemma 4 yields that, for any , sufficiently large , and uniformly for , Note that It is easy to check that , . Moreover, since , we have, for any and . Therefore, we have, as , uniformly for , Furthermore, by the definition of , for sufficiently small and sufficiently large , similarly as (36), it yields that Substituting (41) and (42) into (39) and letting , (38) holds and Theorem 9 is proved in the case of .
Now suppose (29) holds for ; for the case of , using the similar argument as (32), it holds that To estimate and , similarly as (33) and (34), by Lemma 1, Lemma 4, and induction hypothesis, we have, for any , all , sufficiently small , as , Finally for , similarly as (35), we arrive at Combining (43)–(45) and letting , we have To obtain the reverse inequality, for any and , by Lemma 4 and induction hypothesis, it holds that, for any , sufficiently small , Letting , we get Therefore, Theorem 9 follows from (46) and (48) immediately.