Consider the compound renewal risk model where the claim sizes
\(\{X_{ij},j\geq 1\}\) caused by the
ith (
\(i\geq 1\)) accident form a sequence of nonnegative random variables (r.v.s) with finite mean. The interarrival times of the accidents
\(\{\theta _{i},i\geq 1\}\) are positive independent identically distributed (i.i.d.) r.v.s with finite mean
\(\lambda ^{-1}\). Then the renewal counting process is
$$ N(t)=\sup \Biggl\{ n\geq 1: \tau _{n}=\sum _{i=1}^{n}\theta _{i}\leq t \Biggr\} ,\quad t\geq 0, $$
with a mean function
\(\lambda (t)=EN(t)\),
\(t\geq 0\), such that
\(\lambda (t)/\lambda t\to 1\) as
\(t\to \infty \). Let
\(\{Y_{i},i\geq 1 \}\) be the claim numbers caused by the successive accidents, which are a sequence of i.i.d. positive integer-valued r.v.s with finite mean
ν. We assume that the r.v.s
\(\{Y_{i},i\geq 1\}\) are bounded, that is, there exists a finite integer number
\(h>0\) such that
\(Y_{i}\leq h\),
\(i\geq 1\). Thus the aggregate claims accumulated up to time
\(t\geq 0\) are expressed as
$$\begin{aligned} S(t)=\sum_{i=1}^{N(t)}\sum _{j=1}^{Y_{i}}X_{ij}. \end{aligned}$$
(1.1)
In this paper, we investigate the precise large deviations for the random sums
\(S(t)\).
1.1 Heavy-tailed distribution classes
In this subsection, we introduce some related heavy-tailed distribution classes. For a proper distribution
V on
\((-\infty , \infty )\), let
\(\overline{V}=1-V\) be its (right) tail. A distribution
V on
\((-\infty , \infty )\) is said to be heavy-tailed if
$$\begin{aligned} \int _{-\infty }^{\infty }e^{\lambda x}V(dx)=\infty \quad \text{for all } \lambda >0, \end{aligned}$$
that is, if
V has no any positive exponential moment. Otherwise,
V is said to be light-tailed.
The dominated variation distribution class is an important class of heavy-tailed distributions, denoted by
\(\mathscr{D}\). A distribution
V on
\((-\infty , \infty )\) belongs to the class
\(\mathscr{D}\) if for any
\(y\in (0,1)\),
$$ \limsup_{x\to \infty }\frac{\overline{V}(xy)}{\overline{V}(x)}< \infty . $$
A slightly smaller class is the consistent variation distribution class, denoted by
\(\mathscr{C}\). A distribution
V on
\((-\infty , \infty )\) belongs to the class
\(\mathscr{C}\) if
$$ \lim_{y\nearrow 1}\limsup_{x\to \infty } \frac{{\overline{V}}(xy)}{ {\overline{V}}(x)}=1\quad \text{or, equivalently,}\quad \lim _{y\searrow 1}\liminf_{x\to \infty }\frac{{\overline{V}}(xy)}{{\overline{V}}(x)}=1. $$
Another related distribution class is the long-tailed distribution class, denoted by
\(\mathscr{L}\). A distribution
V on
\((-\infty , \infty )\) belongs to the class
\(\mathscr{L}\) if for any
\(y>0\),
$$ \lim_{x\to \infty }\frac{\overline{V}(x+y)}{\overline{V}(x)}=1. $$
A special distribution class is called the extended regularly varying tailed distribution class, denoted by
ERV. A distribution
V on
\((-\infty , \infty )\) belongs to the class
\(\mathit{ERV}(- \alpha ,-\beta )\) for
\(0\leq \alpha \leq \beta <\infty \) if for any
\(y>1\),
$$ y^{-\beta }\leq \liminf_{x\to \infty }\frac{\overline{V}(xy)}{ \overline{V}(x)} \leq \limsup_{x\to \infty }\frac{\overline{V}(xy)}{ \overline{V}(x)}\leq y^{-\alpha }. $$
It is well known that the following relationships hold:
$$ \mathit{ERV}\subset \mathscr{C}\subset \mathscr{L}\cap \mathscr{D} \subset \mathscr{D}. $$
See, for example, Embrechts et al. [
9], Foss et al. [
10], and Denisov et al. [
7]. For a distribution
V on
\((-\infty , \infty )\), denote its upper Matuszewska index by
$$\begin{aligned} J^{+}_{V}=-\lim_{y\to \infty } \frac{\log {\overline{V}}_{*}(y)}{ \log y},\quad \text{with } {\overline{V}}_{*}(y):= \liminf_{x\to \infty }\frac{{\overline{V}}(xy)}{{\overline{V}}(x)}, \quad \text{for } y>1. \end{aligned}$$
Another important parameter
\(L_{V}\) is defined by
$$ L_{V}=\lim_{y\searrow 1}{\overline{V}}_{*}(y). $$
From Chap. 2.1 of Bingham et al. [
2] we get the following equivalent statements:
$$ (\mathrm{i})\quad V\in \mathscr{D};\qquad (\mathrm{ii})\quad 0< L_{V}\leq 1; \qquad (\mathrm{iii})\quad J^{+}_{V}< \infty . $$
1.2 Dependence structures
In the studies of dependent risk models, many works consider a general dependence structure, namely upper tail asymptotically independent (UTAI).
The UTAI structure was proposed by Geluk and Tang [
12] when they investigated the asymptotics of the tail of sums with dependent increments. There are some papers investigating the UTAI structure, such as Asimit et al. [
1], Liu et al. [
22], Gao and Liu [
11], Li [
20], and so on. In this paper, we consider a related dependence structure with the UTAI structure, which was introduced by He et al. [
14] as follows:
$$\begin{aligned} \lim_{n\to \infty }\sup_{x\geq \alpha n}\sup _{1\leq i< j\leq n}xP(\xi _{i}>x|\xi _{j}>x)=0 \quad \text{for all } \alpha >0. \end{aligned}$$
(1.2)
Another dependence structure is the widely (upper/lower) orthant dependent (WUOD/ WLOD) r.v.s, which was introduced by Wang et al. [
29] when they investigated a dependent risk model.
Obviously, if r.v.s
\(\{X_{i},i\geq 1\}\) are WUOD or satisfy relation (
1.2), then r.v.s
\(\{X_{i},i\geq 1\}\) are following the UTAI structure. In addition, Liu et al. [
22] noted that there exist random variables that are UTAI but do not satisfy WUOD structure; see Example 3.1 of Liu et al. [
22]. He et al. [
14] pointed out that if
\(\{X_{i},i\geq 1 \}\) are identically distributed and WUOD random variables with finite means, then
\(\{X_{i},i\geq 1\}\) satisfy (
1.2).
In this paper, when investigating the asymptotic upper bound of the precise large deviations of the aggregate claims, we also consider the claim sizes that have a pairwise negatively quadrant dependence structure. This structure is stronger than the upper tail asymptotically independence structure.
The negatively quadrant dependence structure was introduced by Lehmann [
19]. Many researchers have studied the negatively quadrant dependence structure, such as Ebrahimi and Ghosh [
8], Block et al. [
3], Chen and Ng [
4], Yang et al. [
32], and so on.
For the dependence structure between the interarrival times of accidents and the claim numbers, Liu et al. [
21] adopted a general dependence structure via the conditional tail probability of the accident interarrival time given the claim number caused by the subsequent accidents being fixed, which was based on Chen and Yuen [
5]. In the following, we briefly restate the definition of the above dependence structure between the interarrival times of accidents and the claim numbers and some other related quantities.
Assume that
\(\{(\theta _{i}, Y_{i}),i\geq 1\}\) are i.i.d. We denote by
\((\theta , Y)\) the generic r.v. of
\(\{(\theta _{i}, Y_{i}),i\geq 1\}\) and assume that
θ and
Y are such that for all
\(t\in [0, \infty )\) and any
\(1\leq k\leq h\),
$$\begin{aligned} P(\theta >t|Y=k)\leq P\bigl(\theta ^{*}>t\bigr), \end{aligned}$$
(1.3)
where
\(\theta ^{*}\) is a nonnegative r.v. independent of other sources of randomness.
Let
\(\theta _{1}^{*}\) be a positive r.v. independent of all sources of randomness and such that for all
\(t\in [0,\infty )\) and r any
\(1\leq k\leq h\),
\(P(\theta _{1}^{*}>t)=P(\theta _{1}>t|Y_{1}=k)\). Denote
\(\tau _{1}^{*}=\theta _{1}^{*}\),
\(\tau _{n}^{*}=\theta _{1}^{*}+\sum_{i=2} ^{n}\theta _{i}\),
\(n\geq 2\), which constitute the delayed renewal counting process
$$\begin{aligned} N^{*}(t)=\sup \bigl\{ n\geq 1: \tau _{n}^{*} \leq t\bigr\} ,\quad t\geq 0. \end{aligned}$$
(1.4)
Then, for all integer
\(n\geq 1\) and
\(1\leq k\leq h\),
$$\begin{aligned} P\bigl(N^{*}(t)=n\bigr) =& P\bigl(\tau _{n}^{*} \leq t, \tau _{n+1}^{*}>t\bigr) \\ =& \int _{0}^{\infty }P \Biggl(\sum _{i=2}^{n}\theta _{i}\leq t-u, \sum _{i=2}^{n+1}\theta _{i}> t-u \Biggr) P\bigl(\theta _{1}^{*}\in du\bigr) \\ =& \int _{0}^{\infty }P \Biggl(\sum _{i=2}^{n}\theta _{i}\leq t-u, \sum _{i=2}^{n+1}\theta _{i}> t-u \Biggr) P(\theta _{1}\in du|Y _{1}=k) \\ =& P \Biggl(\sum_{i=1}^{n}\theta _{i}\leq t, \sum_{i=1} ^{n+1}\theta _{i}> t\Big|Y_{1}=k \Biggr) \\ =& P\bigl(N(t)=n|Y_{1}=k\bigr). \end{aligned}$$
(1.5)
1.3 Motivation and main results
It is well known that for the standard renewal risk model, when the claims sizes and interarrival times are all i.i.d. r.v.s, the precise large deviations for the aggregate claims and the ruin probability have been widely studied; see Klüppelberg and Stadtmüller [
15], Tang [
25], Wang [
28], and Hao and Tang [
13], among others. Actually, the independence assumption does not apply to most practical problems. Based on this situation, many researchers started to pay their attention to the risk model with dependent assumptions; we refer to Chen and Ng [
4], Konstantinides and Loukissas [
18], Wang et al. [
29], Peng and Wang [
23], Peng and Wang [
24], Yang et al. [
33], and some others.
There are many works investigating the compound renewal risk model. When the claim sizes are i.i.d. r.v.s with common distribution
F, Tang et al. [
26] investigated the large deviations for the aggregate claims with
\(F\in \mathit{ERV}\); Konstantinides and Loukissas [
17] also considered the i.i.d. claim sizes and extended the results of Tang et al. [
26] from
\(F\in \mathit{ERV}\) to
\(F\in \mathscr{C}\); Zong [
34] established an asymptotic formula for the finite-time ruin probability of compound renewal risk model in which claims sizes and interarrival times are all identically distributed but negatively dependent; Chen et al. [
6] considered the claim sizes that are a sequence of negatively dependent heavy-tailed r.v.s with common distribution
\(F\in \mathscr{C}\); Yang and Wang [
31] investigated the precise large deviations for dependent random variables and assumed that the claim sizes are extended negatively dependent r.v.s with common distribution
\(F\in \mathscr{D}\).
In the researches mentioned, the assumption of independence oft the interarrival times of the accidents and the corresponding claim numbers was always used. Recently, Liu et al. [
21] considered the dependent case (
1.3) between the interarrival times of the accidents and the corresponding claim numbers. They investigated the case where
\(\{X_{ij}, i\geq 1, j\geq 1\}\) are nonnegative r.v.s with common distribution
F,
\(\{\theta _{i}, i \geq 1\}\) and
\(\{Y_{i}, i\geq 1\}\) are all i.i.d. r.v.s, but for every
\(i\geq 1\),
\(\theta _{i}\) and
\(Y_{i}\) follow the dependence structure (
1.3). Liu et al. [
21] obtained the asymptotic lower bound of the precise large deviations of the aggregate claims when
\(F\in \mathscr{L}\cap \mathscr{D}\) and
\(\{X_{ij},i\geq 1,j\geq 1\}\) satisfy the dependence structure (
1.2). Strengthening the condition to that
\(F\in \mathscr{C}\) and
\(\{X_{ij},i\geq 1,j\geq 1\}\) are WUOD r.v.s with
\(EX_{1}^{\beta }<\infty \) for some
\(\beta >1\), they derived the asymptotic upper bound of the precise large deviations of the aggregate claims.
In this paper, we still consider the dependent case (
1.3) between the interarrival times of the accidents and the corresponding claim numbers and investigate the precise large deviations of the aggregate claims (
1.1). We mainly study the case where the claim sizes caused by the different accidents are dependent and have different distributions belonging to the dominated variation distribution class. To proceed, we impose the following assumptions.
Firstly, we investigate the asymptotic lower bound of the precise large deviations for the aggregate claims. For this case, we will use the following assumption on the distributions \(F_{i}\), \(i\geq 1\).
From Assumption
1.3 we know that if
\(F_{i}\in \mathscr{D}\),
\(i\geq 1\), then
\(F\in \mathscr{D}\), and for any
\(y>1\) and
\(i\geq 1\),
$$\begin{aligned} \frac{M_{L}}{M_{U}}\overline{F_{i*}}(y)\leq \overline{F_{*}}(y) \leq \frac{M _{U}}{M_{L}}\overline{F_{i*}}(y). \end{aligned}$$
Therefore we have that
\(J_{F}^{+}=J_{F_{i}}^{+}\) and
\(\frac{M_{L}}{M _{U}}L_{F_{i}}\leq L_{F}\leq \frac{M_{U}}{M_{L}}L_{F_{i}}\) for all
\(i\geq 1\). Under Assumptions
1.1,
1.2, and
1.3, we can obtain the asymptotic lower bound of the precise large deviations of the aggregate claims.
When
\(\{X_{ij},i\geq 1,j\geq 1\}\) are identically distributed r.v.s, we obtain the following corollary directly from Theorem
1.1.
Next, we will investigate the asymptotic upper bound of the precise large deviations for the aggregate claims. In this case, we consider the case where the claim sizes \(\{X_{ij},i\geq 1,j\geq 1\}\) follow the pairwise negatively quadrant dependence structure.
Furthermore, we will use the following assumption, which was given by Yang and Wang [
31] when they investigated the precise large deviations for extendedly negatively dependent random variables. Wang et al. [
30] also used this assumption when they studied the precise large deviations for widely orthant dependent random variables.
As noted in Remark 1(ii) of Wang et al. [
30] Assumption
1.5 actually requires the distributions of
\(X_{ij}\),
\(i\geq 1\),
\(j\geq 1\), do not differ too much from each other. In particular, if there exists a positive integer
\(i_{0}\) such that
\(F_{i}=F_{i_{0}}\) for all
\(i\geq i_{0}\), then since
\(F_{i_{0}}\in \mathscr{D}\), we know that Assumption
1.5 is satisfied.
We will derive the asymptotic upper bound of the precise large deviations of the aggregate claims under Assumptions
1.2,
1.3,
1.4, and
1.5.
If
\(\{X_{ij},i\geq 1,j\geq 1\}\) are identically distributed r.v.s with common distribution
\(F\in \mathscr{D}\), then Assumptions
1.3 and
1.5 are satisfied. Thus from Theorem
1.2 we can obtain the following corollary.
The rest of the paper is organized as follows. Section
2 includes some lemmas. In Sect.
3, we collect the proofs of our main results.