# Precise Large Deviations for Random Sums of END Random Variables with Dominated Variation

Authors: Yu Chen; Zhihui Qu
Journal: ISRN Applied Mathematics
 (2013)

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## Abstract

We investigate the precise large deviations for random sums of extended negatively dependent random variables with long and dominatedly varying tails. We find out that the asymptotic behavior of precise large deviations of random sums is insensitive to the extended negative dependence. We apply the results to a generalized dependent compound renewal risk model including premium process and claim process and obtain the asymptotic behavior of the tail probabilities of the claim surplus process.


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## Body

## 1. Introduction

The study of large deviations plays an important role in insurance and finance theory. The aim of this paper is to study precise large deviation probabilities for sequences of dependent and heavy-tailed random variables. Let{Xk,k≥1} be a sequence of real-valued heavy-tailed random variables with common distribution function F and finite mean μ. We denote its tail by F¯=1-F. We say that X (or its distribution F) is heavy-tailed if it has no exponential moments. Suppose that {N(t),t≥0} is a nonnegative integer-valued process independent of the sequence {Xk,k≥1} with mean function λ(t)=EN(t) which tends to ∞ as t→∞. For random sums of the type
(1)SN(t)=∑k=1N(t)‍Xk,t≥0,(∑k=10‍Xk:=0),
we aim to study the precise large deviation probabilities
(2)P(SN(t)-ESN(t)>x)=P(SN(t)-μλ(t)>x).Some earlier work, for the case{Xk,k≥1} are independent, we refer the reader to see Cline and Hsing [1], Klüppelberg and Mikosch [2], Tang et al. [3], Ng et al. [4], Konstantinides and Loukissas [5] and Loukissas [6], and so forth; for the case {Xk,k≥1} are negatively dependent, see Chen and Zhang [7], Tang [8], Konstantinides and Loukissas [5], Chen et al. [9] and Wang et al. [10], and so forth.One of the main concepts we use is the extended negative dependence, which was first introduced by Liu [11] and Chen et al. [9]. The definition of this dependence structure is given in the following.Definition 1.
We call random variables{Xk,k≥1} extended negatively dependent (END), if there exists a constant M>0 such that
(3)P(X1≤x1,…,Xn≤xn)≤M∏k=1n‍P(Xk≤xk),(4)P(X1>x1,…,Xn>xn)≤M∏k=1n‍P(Xk>xk)
hold for each n≥1 and all x1,…,xn.Recall that ifM=1 for any integer n≥1 in (3) and (4), then random variables {Xk,k≥1} are called lower negatively dependent (LND) and upper negatively dependent (UND), respectively. {Xk,k≥1} are called negatively dependent (ND) if both (3) and (4) hold, so an ND sequence must be an END sequence. From Liu [11], the END structure can reflect a negative dependence structure but also a positive one to some extent, so the END structure is substantially more comprehensive than the ND structure.In presence of END structure, Liu [11] obtained that the precise large deviations of partial sums with consistent variation are insensitive to the dependence structure. Chen et al. [9] extended the previous results of Liu [11] to random sums with consistently varying tails. Wang et al. [10] considered a wider dependent structure and investigated the precise large deviations for the partial sums with dominatedly varying tails.The basic assumption of this paper is that{Xk,k≥1} is a sequence of real-valued END random variables with common heavy-tailed distribution F and finite mean μ. We consider a wider heavy-tailed distribution class than consistently varying class, the intersection of long-tailed class, and dominatedly varying-tailed class. The main aim of this paper is to extend the study to certain END cases and find out whether the asymptotic behavior of precise large deviations is insensitive to the extended negative dependence.The rest of the paper is organized as follows. After simply reviewing some subclasses of heavy-tailed distributions and giving some lemmas needed to prove the theorem in Section2, we derive our main result of precise large deviations for random sums in Section 3 and apply the result to the dependent compound renewal risk model in Section 4.


## 2. Preliminaries

### 2.1. Heavy-Tailed Distribution Classes

To formulate our main results we need to introduce some notations and assumptions. Throughout this paper we restrict ourselves to the case that{Xk,k≥1} is a sequence of heavy-tailed random variables. In risk theory, heavy-tailed distributions are often used to model large claim amounts. They play a key role in some fields such as insurance, financial mathematics, and queueing theory. Next, we recall some important subclasses of heavy-tailed distributions.A distribution function (d.f.)F is said to belong to class 𝒟, if
(5)limsupx→∞F¯(xy)F¯(x)<∞
holds for some 0<y<1; denote that F∈𝒟. Such a d.f. F is said to have a dominatedly varying tail.A d.f.F is said to belong to class ℒ, if
(6)limx→∞F¯(x+y)F¯(x)=1
holds for all y∈(-∞,+∞); denote that F∈ℒ. Such a d.f. F is said to have a long tail.A d.f.F is said to belong to class 𝒞, if
(7)limy↘1liminfx→∞F¯(xy)F¯(x)=1,orequivalently,limy↗1limsupx→∞F¯(xy)F¯(x)=1.
Denote that F∈𝒞. Now the d.f. F is said to have a consistently varying tail.It is well known that the previous heavy-tailed distribution classes have the following relationship:(8)𝒞⊂ℒ∩𝒟⊂ℒ.For more details about heavy-tailed class in the context of insurance and finance, we refer the reader to Embrechts et al. [12], Rolski et al. [13], and Tang and Tsitsiashvili [14], among others.For a d.f.F, we define
(9)𝕁F=inf{-logF¯*(y)logy,y>1}withF¯*(y)=liminfx→∞F¯(xy)F¯(x)fory>1,
and LF=limy↓1F¯*(y); we call 𝕁F the (upper) Matuszewska index of the d.f. F. From Bingham et al. [15, Chapter 2.1], we know that the following assertions are equivalent:
(10)(i)F∈𝒟;(ii)0<LF≤1;(iii)𝕁F<∞.
From the definition of the class 𝒞, it holds that F∈𝒞 if and only if LF=1. Denote that
(11)γF(y)=liminfx→∞F¯(x+y)F¯(x),y≥0.
If F∈ℒ, we know that γF(y)=1.We close this section by explaining some symbols which will be used later. We will use≲,≳,~, and ≍  to connect two positive functions, say f1(x) and f2(x), as follows:
(12)f1(x)≍f2(x)if0<liminfx→∞f1(x)f2(x)f1(x)≍f2(x)if0≤limsupx→∞f1(x)f2(x)<∞;f1(x)≲f2(x)iflimsupx→∞f1(x)f2(x)≤1;f1(x)≳f2(x)ifliminfx→∞f1(x)f2(x)≥1;f1(x)~f2(x)iflimx→∞f1(x)f2(x)=1.
We say that f1(x) and f2(x) are weakly equivalent if f1(x)≍f2(x). Throughout, every limit relation without explicit limit is with respect to t→∞.


### 2.2. Some Lemmas

We will need some lemmas used in the proofs of our theorems. The following lemma is given by Tang [8].Lemma 2.
IfF∈𝒟, then(1)
for eachp>𝕁F, there exist positive constants C and D such that the inequality
(13)F¯(x)F¯(y)≤C(xy)p
holds for all x≥y≥D;(2)
it holds for eachp>𝕁F that x-p=o(F¯(x)).Lemma 3.
IfF∈𝒟, then for any fixed θ>0,a(t) satisfying a(t)→∞ast→∞,
(14)LFF¯(x)≲F¯(x+o(a(t)))≲LF-1F¯(x)
holds uniformly for x≥θa(t).Proof.
The proof is analogous to that of Lemma 2.2 of Chen and Zhang [7] with some minor modifications and is omitted.By Definition1, the following properties of END sequences can be obtained directly.Lemma 4.
Let{Xk,k≥1} be a sequence of END random variables, and then(1)
{fk(Xk),k≥1} are still END, where {fk(·),k≥1} are either all monotone increasing or all monotone decreasing;(2)
for anyn=1,2,…, there exists a constant M>0 such that
(15)E(∏k=1n‍Xk+)≤M∏k=1n‍EXk+.Wang et al. [10] proved the following result for partial sums.Lemma 5.
Let{Xk,k≥1} be a sequence of END random variables with common distribution F∈𝒟 and finite mean μ=0, satisfying
(16)F(-x)=o(F¯(x))asx→∞,E|X1|rI{X1≤0}<∞forsomer>1.
Then, for any γ>0,
(17)LF≤liminfn→∞infx≥γnP(Sn>x)nF¯(x)≤limsupn→∞supx≥γnP(Sn>x)nF¯(x)≤LF-1.Lemma 6.
Let{Xk,k≥1} be a sequence of END random variables with common distribution F∈𝒟 and finite mean μ, satisfying
(18)F(-x)=o(F¯(x))asx→∞,E|X1|rI{X1≤0}<∞forsomer>1.
Then, for any γ>0,
(19)LFγF(|μ|)≤liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x)≤limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x)≤LF-1γF-2(-μ∧0).Proof.
LetXk′=Xk-μ,k≥1, and denote its distribution by F′. It is easy to see that F′(x)=F(x+μ) and
(20)𝕀F(y)=liminfx→∞F¯(x+y)F¯(x)=liminfx→∞F¯(x+μ+y)F¯(x+μ)=γF′(y).
(i)  Considerμ>0.
ByF∈𝒟 and F′(x)=F(x+μ), it is easy to see that F′(x)∈𝒟. In fact for any 0<y<1,
(21)limsupx→∞F¯′(xy)F¯′(x)=limsupx→∞F¯(xy+μ)F¯(x+μ)≤limsupx→∞F¯((x+μ)y)F¯(x+μ)<∞.
By the definition γF(y), we see that
(22)LF′(y)=limy↓1liminfx→∞F¯(xy+μ)F¯(x+μ)≥limy↓1liminfx→∞F¯((x+μ)y)F¯(x+μ)=LF(y).
Thus, by Lemma 5, we have
(23)limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x)=limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x+μ)F¯(x+μ)F¯(x)≤limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x+μ)≤LF′-1≤LF-1,liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x)=liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x+μ)F¯(x+μ)F¯(x)≥liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x+μ)liminfx→∞F¯(x+μ)F¯(x)≥LF′γF(μ)≥LFγF(μ).
(ii) Considerμ<0.
First we shows thatF′(x)∈𝒟. For large x there exist ϵ such that μ>-ϵx. For some y>ϵ and F∈𝒟,
(24)limsupx→∞F¯′(xy)F¯′(x)=limsupx→∞F¯(xy+μ)F¯(x)≤limsupx→∞F¯((y-ϵ)x)F¯(x+μ)<∞.
By the definition γF(y), we see that
(25)LF′(y)=limy↓1liminfx→∞F¯(xy+μ)F¯(x+μ)≥limy↓1liminfx→∞F¯(xy)F¯(x)liminfx→∞F¯(x)F¯(x+μ)=LFγF(-μ).
Similarly, by Lemma 5, we have
(26)limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x)=limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x+μ)F¯(x+μ)F¯(x)≤limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x+μ)limsupn→∞F¯(x+μ)F¯(x)≤LF′-1(liminfx→∞F¯(x)F¯(x+μ))-1=LF-1γF-2(-μ),liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x)=liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x+μ)F¯(x+μ)F¯(x)≥liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x+μ)liminfx→∞F¯(x+μ)F¯(x)≥LF′≥LFγF(-μ).Lemma 7.
Let{Xk,k≥1} be END with common distribution F∈𝒟 and mean 0, satisfying (18). Denote that Sn=∑k=1n‍Xk,n≥1. Then, for each fixed γ>0 and some C=C(γ) irrespective to x and n, the inequality
(27)P(Sn>x)≤CnF¯(x)
holds uniformly for all x≥γn and n=1,2,….Proof.
Firstly, by Lemma5, we know that
(28)P(Sn>x)≲LF-1nF¯(x)
holds uniformly for all x≥γn. Moreover, for each fixed positive integer n0, by Lemma 2, the inequality
(29)P(∑k=1n‍Xk>x)≤P(⋃i=1n‍{Xk>xn})≤nF¯(xn0)≤C(n0)nF¯(x)
holds for some C(n0)>0, all x≥0, and all n=1,2,…,n0. Hence, we complete the proof.Lemma 8.
Let{Xk,k≥1} be a sequence of END random variables with common distribution F and finite mean μ; if 0<μ+=EX1I{X1>0}<∞, then for any v>0 and some C=C(v)>0, the inequality
(30)P(∑k=1n‍Xk>x)≤nP(X1>xv)+C(nx)v
holds for all n=1,2,… and x>0.This lemma plays an important role in the proof of Theorem9, which is Lemma 2.3 of Chen et al. [9].


## 2.1. Heavy-Tailed Distribution Classes

To formulate our main results we need to introduce some notations and assumptions. Throughout this paper we restrict ourselves to the case that{Xk,k≥1} is a sequence of heavy-tailed random variables. In risk theory, heavy-tailed distributions are often used to model large claim amounts. They play a key role in some fields such as insurance, financial mathematics, and queueing theory. Next, we recall some important subclasses of heavy-tailed distributions.A distribution function (d.f.)F is said to belong to class 𝒟, if
(5)limsupx→∞F¯(xy)F¯(x)<∞
holds for some 0<y<1; denote that F∈𝒟. Such a d.f. F is said to have a dominatedly varying tail.A d.f.F is said to belong to class ℒ, if
(6)limx→∞F¯(x+y)F¯(x)=1
holds for all y∈(-∞,+∞); denote that F∈ℒ. Such a d.f. F is said to have a long tail.A d.f.F is said to belong to class 𝒞, if
(7)limy↘1liminfx→∞F¯(xy)F¯(x)=1,orequivalently,limy↗1limsupx→∞F¯(xy)F¯(x)=1.
Denote that F∈𝒞. Now the d.f. F is said to have a consistently varying tail.It is well known that the previous heavy-tailed distribution classes have the following relationship:(8)𝒞⊂ℒ∩𝒟⊂ℒ.For more details about heavy-tailed class in the context of insurance and finance, we refer the reader to Embrechts et al. [12], Rolski et al. [13], and Tang and Tsitsiashvili [14], among others.For a d.f.F, we define
(9)𝕁F=inf{-logF¯*(y)logy,y>1}withF¯*(y)=liminfx→∞F¯(xy)F¯(x)fory>1,
and LF=limy↓1F¯*(y); we call 𝕁F the (upper) Matuszewska index of the d.f. F. From Bingham et al. [15, Chapter 2.1], we know that the following assertions are equivalent:
(10)(i)F∈𝒟;(ii)0<LF≤1;(iii)𝕁F<∞.
From the definition of the class 𝒞, it holds that F∈𝒞 if and only if LF=1. Denote that
(11)γF(y)=liminfx→∞F¯(x+y)F¯(x),y≥0.
If F∈ℒ, we know that γF(y)=1.We close this section by explaining some symbols which will be used later. We will use≲,≳,~, and ≍  to connect two positive functions, say f1(x) and f2(x), as follows:
(12)f1(x)≍f2(x)if0<liminfx→∞f1(x)f2(x)f1(x)≍f2(x)if0≤limsupx→∞f1(x)f2(x)<∞;f1(x)≲f2(x)iflimsupx→∞f1(x)f2(x)≤1;f1(x)≳f2(x)ifliminfx→∞f1(x)f2(x)≥1;f1(x)~f2(x)iflimx→∞f1(x)f2(x)=1.
We say that f1(x) and f2(x) are weakly equivalent if f1(x)≍f2(x). Throughout, every limit relation without explicit limit is with respect to t→∞.


## 2.2. Some Lemmas

We will need some lemmas used in the proofs of our theorems. The following lemma is given by Tang [8].Lemma 2.
IfF∈𝒟, then(1)
for eachp>𝕁F, there exist positive constants C and D such that the inequality
(13)F¯(x)F¯(y)≤C(xy)p
holds for all x≥y≥D;(2)
it holds for eachp>𝕁F that x-p=o(F¯(x)).Lemma 3.
IfF∈𝒟, then for any fixed θ>0,a(t) satisfying a(t)→∞ast→∞,
(14)LFF¯(x)≲F¯(x+o(a(t)))≲LF-1F¯(x)
holds uniformly for x≥θa(t).Proof.
The proof is analogous to that of Lemma 2.2 of Chen and Zhang [7] with some minor modifications and is omitted.By Definition1, the following properties of END sequences can be obtained directly.Lemma 4.
Let{Xk,k≥1} be a sequence of END random variables, and then(1)
{fk(Xk),k≥1} are still END, where {fk(·),k≥1} are either all monotone increasing or all monotone decreasing;(2)
for anyn=1,2,…, there exists a constant M>0 such that
(15)E(∏k=1n‍Xk+)≤M∏k=1n‍EXk+.Wang et al. [10] proved the following result for partial sums.Lemma 5.
Let{Xk,k≥1} be a sequence of END random variables with common distribution F∈𝒟 and finite mean μ=0, satisfying
(16)F(-x)=o(F¯(x))asx→∞,E|X1|rI{X1≤0}<∞forsomer>1.
Then, for any γ>0,
(17)LF≤liminfn→∞infx≥γnP(Sn>x)nF¯(x)≤limsupn→∞supx≥γnP(Sn>x)nF¯(x)≤LF-1.Lemma 6.
Let{Xk,k≥1} be a sequence of END random variables with common distribution F∈𝒟 and finite mean μ, satisfying
(18)F(-x)=o(F¯(x))asx→∞,E|X1|rI{X1≤0}<∞forsomer>1.
Then, for any γ>0,
(19)LFγF(|μ|)≤liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x)≤limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x)≤LF-1γF-2(-μ∧0).Proof.
LetXk′=Xk-μ,k≥1, and denote its distribution by F′. It is easy to see that F′(x)=F(x+μ) and
(20)𝕀F(y)=liminfx→∞F¯(x+y)F¯(x)=liminfx→∞F¯(x+μ+y)F¯(x+μ)=γF′(y).
(i)  Considerμ>0.
ByF∈𝒟 and F′(x)=F(x+μ), it is easy to see that F′(x)∈𝒟. In fact for any 0<y<1,
(21)limsupx→∞F¯′(xy)F¯′(x)=limsupx→∞F¯(xy+μ)F¯(x+μ)≤limsupx→∞F¯((x+μ)y)F¯(x+μ)<∞.
By the definition γF(y), we see that
(22)LF′(y)=limy↓1liminfx→∞F¯(xy+μ)F¯(x+μ)≥limy↓1liminfx→∞F¯((x+μ)y)F¯(x+μ)=LF(y).
Thus, by Lemma 5, we have
(23)limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x)=limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x+μ)F¯(x+μ)F¯(x)≤limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x+μ)≤LF′-1≤LF-1,liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x)=liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x+μ)F¯(x+μ)F¯(x)≥liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x+μ)liminfx→∞F¯(x+μ)F¯(x)≥LF′γF(μ)≥LFγF(μ).
(ii) Considerμ<0.
First we shows thatF′(x)∈𝒟. For large x there exist ϵ such that μ>-ϵx. For some y>ϵ and F∈𝒟,
(24)limsupx→∞F¯′(xy)F¯′(x)=limsupx→∞F¯(xy+μ)F¯(x)≤limsupx→∞F¯((y-ϵ)x)F¯(x+μ)<∞.
By the definition γF(y), we see that
(25)LF′(y)=limy↓1liminfx→∞F¯(xy+μ)F¯(x+μ)≥limy↓1liminfx→∞F¯(xy)F¯(x)liminfx→∞F¯(x)F¯(x+μ)=LFγF(-μ).
Similarly, by Lemma 5, we have
(26)limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x)=limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x+μ)F¯(x+μ)F¯(x)≤limsupn→∞supx≥γnP(Sn-nμ>x)nF¯(x+μ)limsupn→∞F¯(x+μ)F¯(x)≤LF′-1(liminfx→∞F¯(x)F¯(x+μ))-1=LF-1γF-2(-μ),liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x)=liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x+μ)F¯(x+μ)F¯(x)≥liminfn→∞infx≥γnP(Sn-nμ>x)nF¯(x+μ)liminfx→∞F¯(x+μ)F¯(x)≥LF′≥LFγF(-μ).Lemma 7.
Let{Xk,k≥1} be END with common distribution F∈𝒟 and mean 0, satisfying (18). Denote that Sn=∑k=1n‍Xk,n≥1. Then, for each fixed γ>0 and some C=C(γ) irrespective to x and n, the inequality
(27)P(Sn>x)≤CnF¯(x)
holds uniformly for all x≥γn and n=1,2,….Proof.
Firstly, by Lemma5, we know that
(28)P(Sn>x)≲LF-1nF¯(x)
holds uniformly for all x≥γn. Moreover, for each fixed positive integer n0, by Lemma 2, the inequality
(29)P(∑k=1n‍Xk>x)≤P(⋃i=1n‍{Xk>xn})≤nF¯(xn0)≤C(n0)nF¯(x)
holds for some C(n0)>0, all x≥0, and all n=1,2,…,n0. Hence, we complete the proof.Lemma 8.
Let{Xk,k≥1} be a sequence of END random variables with common distribution F and finite mean μ; if 0<μ+=EX1I{X1>0}<∞, then for any v>0 and some C=C(v)>0, the inequality
(30)P(∑k=1n‍Xk>x)≤nP(X1>xv)+C(nx)v
holds for all n=1,2,… and x>0.This lemma plays an important role in the proof of Theorem9, which is Lemma 2.3 of Chen et al. [9].


## 3. Precise Large Deviations for Random Sums

### 3.1. Main Theorem

Theorem 9.
Let{Xk,k≥1} be a sequence of real-valued END random variables with common distribution F∈ℒ∩𝒟 and finite mean μ, satisfying (18). Let {N(t),t≥0} be a nonnegative integer-valued process independent of {Xk,k≥1}, and assume that N(t) satisfies
(31)AssumptionN1:EN(t)pI(N(t)>(1+δ)λ(t))=O(λ(t))
for some p>𝕁F and
(32)AssumptionN2:P(N(t)≤(1-δ)λ(t))=o(λ(t)F¯(λ(t)))
for all 0<δ<1. SN(t) is defined by (1), and then for any γ>0,
(33)LF2λ(t)F¯(x)≲P(SN(t)-μλ(t)>x)≲LF-2λ(t)F¯(x)
holds uniformly for x≥γλ(t).Remark 10.
From the following proof, it is easy to see that (33) holds uniformly for x≥γλ(t) and γ>|μ| if N(t) just satisfies Assumption N1.Corollary 11.
Under the assumptions of Theorem9 and if F∈𝒞, then for any γ>0,
(34)P(SN(t)-μλ(t)>x)~λ(t)F¯(x)
holds uniformly for x≥γλ(t).From Remark10, if μ>0, Corollary 11 is just the same as Theorem 3.1 of Chen et al. [9]. The previous results extend Chen et al. [9] and Chen and Zhang [7].


### 3.2. Proof of Theorem9

In the sequel,C always represents an absolute positive constant, which may vary from different places.In order to prove Theorem9, we divide the random sum into three parts as follows:
(35)P(SN(t)-μλ(t)>x)=∑n=1∞‍P(Sn-μλ(t)>x)P(N(t)=n)=(∑n<(1-δ)λ(t)‍+∑|n-λ(t)|≤δλ(t)‍+∑n>(1+δ)λ(t)‍)×P(Sn-μλ(t)>x)P(N(t)=n),
where δ∈(0,1) is an arbitrarily fixed number to be specified later.By Tang et al. [3], Assumption N1 implies that
(36)N(t)λ(t)→p1;
then by the dominated convergence theorem,
(37)limt→∞E[N(t)λ(t)I(|N(t)-λ(t)|≤δλ(t))]=1.
Therefore, for any 0<δ<1,
(38)E[N(t)I(N(t)>(1+δ)λ(t))]=o(λ(t)),(39)E[N(t)I(N(t)<(1-δ)λ(t))]=o(λ(t)).Therefore, Theorem9 originates from the following three lemmas.Lemma 12.
Assume that AssumptionsN1 and N2 hold; then for any γ>0,
(40)J1:=∑n<(1-δ)λ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)=o(λ(t)F¯(x))
holds uniformly for x≥γλ(t).Proof.
In order to prove this lemma, we consider the following three cases.
(i)μ≥0 and γ>0.
Choose0<δ<1 such that δμ<γ. Note that n<(1-δ)λ(t), x≥γλ(t)>(γ/(1-δ))n. By Lemma 7,
(41)J1=∑n<(1-δ)λ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)=∑n<(1-δ)λ(t)‍P(Sn-nμ>x+μ(λ(t)-n))P(N(t)=n)≤∑n<(1-δ)λ(t)‍P(Sn-nμ>x)P(N(t)=n)≤CF¯(x)∑n<(1-δ)λ(t)‍nP(N(t)=n)=CF¯(x)EN(t)I(N(t)<(1-δ)λ(t))=o(λ(t)F¯(x)),
and the last equality holds by relation (39).
(ii)μ<0 and γ>|μ|.
Chooseδ such that (γ+μ)/(1-δ)-μ>0. Note that x+μλ(t)-nμ≥((γ+μ)/(1-δ)-μ)n. Mimicking the proof of (i) and by Lemma 7, we have
(42)J1≤∑n<(1-δ)λ(t)‍CnF¯(x+μλ(t)-nμ)P(N(t)=n)≤CF¯(x+μλ(t))EN(t)I(N(t)<(1-δ)λ(t))=o(λ(t)F¯(x)),
where in the last step, we used the relation F¯(x+μλ(t))≤F¯((1+μ/γ)x)≍F¯(x).
(iii)μ<0 and γ∈(0,|μ|].
Letγ1>|μ|. For x≥γ1λ(t), we have x+μλ(t)-nμ>|μ|n. By Lemma 7,
(43)J1≤∑n<(1-δ)λ(t)‍CnF¯(x+μλ(t)-nμ)P(N(t)=n)≤CF¯(x+μλ(t))∑n<(1-δ)λ(t)‍nP(N(t)=n)=CF¯(x+μλ(t))EN(t)I(N(t)<(1-δ)λ(t))=o(λ(t)F¯(x)),
where the last step can be verified as F¯(x+μλ(t))≤F¯((1+μ/γ1)x)≍F¯(x).
Forγλ(t)≤x<γ1λ(t), note that F¯(x)≥F¯(γ1λ(t))≍F¯(λ(t)). Hence, by Assumption N2, the inequality
(44)J1≤P(N(t)≤(1-δ)λ(t))=o(1)λ(t)F¯(λ(t))=o(λ(t)F¯(x))
still holds uniformly for all γλ(t)≤x<γ1λ(t). As a result, the relation
(45)J1=o(λ(t)F¯(x))
holds uniformly for all x≥γλ(t).Lemma 13.
Assume that AssumptionsN1 and N2 hold; then for any γ>0,
(46)LF2λ(t)F¯(x)≲J2:=∑|n-λ(t)|≤δλ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)≲LF-2λ(t)F¯(x)
holds uniformly for x≥γλ(t).Proof.
Similar to the proof of Lemma12, we split into three cases.
(i)μ≥0 and γ>0.
Firstly, for|n-λ(t)|≤δλ(t), then λ(t)-n≥-δλ(t). For x≥γλ(t),
(47)J2=∑|n-λ(t)|≤δλ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)=∑|n-λ(t)|≤δλ(t)‍P(Sn-nμ>x+μ(λ(t)-n))P(N(t)=n)≤∑|n-λ(t)|≤δλ(t)‍P(Sn-nμ>x-μδλ(t))P(N(t)=n)≤∑|n-λ(t)|≤δλ(t)‍P(Sn-nμ>(1-μδγ)x)P(N(t)=n).
By Lemma 6, for any ϵ>0 and sufficient large t,x≥γλ(t)≥(γ/(1+δ))n,
(48)P(Sn-nμ>(1-μδγ)x)≤(1+ϵ)LF-1nF¯((1-μδγ)x).
Then,
(49)J2≤(1+ϵ)LF-1F¯((1-μδγ)x)∑|n-λ(t)|≤δλ(t)‍nP(N(t)=n)=(1+ϵ)LF-1F¯((1-μδγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t));
thus by F∈ℒ∩𝒟, letting δ→0, we have limδ→0F¯((1-μδ/γ)x)/F¯(x)=LF-1. Combining this and (37),
(50)limsupt→∞supx≥γλ(t)J2λ(t)F¯(x)≤LF-2.
On the other hand, by the similar way, for |n-λ(t)|≤δλ(t), then λ(t)-n≤δλ(t),
(51)J2=∑|n-λ(t)|≤δλ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)≥∑|n-λ(t)|≤δλ(t)‍P(Sn-nμ>x+μδλ(t))P(N(t)=n)≥∑|n-λ(t)|≤δλ(t)‍P(Sn-nμ>(1+μδγ)x)P(N(t)=n)≥(1-ϵ)LFγF(μ)F¯((1+δμγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t)).
We have
(52)liminft→∞infx>γλ(t)J2λ(t)F¯(x)≥LF2γF(μ).
Combining (50) and (52), we complete the proof of the lemma in this case.
(ii)μ<0 and γ>|μ|.
Similarly, for anyϵ>0 and sufficient t,x≥γλ(t),
(53)J2≤∑|n-λ(t)|≤δλ(t)‍(1+ϵ)LF-1nF¯(x+μλ(t)-nμ)P(N(t)=n)≤(1+ϵ)LF-1F¯(x+μλ(t)-μ(1-δ)λ(t))×∑|n-λ(t)|≤δλ(t)‍nP(N(t)=n)≤(1+ϵ)LF-1γF-2(-μ)F¯((1+δμγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t)),J2≥(1-ϵ)LFγF(-μ)F¯((1-δμγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t)).
Analogously, we can obtain
(54)limsupt→∞supx≥γλ(t)J2λ(t)F¯(x)≤LF-2γF-2(-μ),liminft→∞infx>γλ(t)J2λ(t)F¯(x)≥LF2γF(-μ).
(iii)μ<0 and γ∈(0,|μ|].
Sinceγ∈(0,|μ|], note that x+μλ(t)-nμ≥(γ+μ)λ(t)-μn≥((γ+μ)/(1-δ)-μ)n, and it follows from Lemma 6 that for all x≥γλ(t),
(55)J2≤∑|n-λ(t)|≤δλ(t)‍(1+ϵ)LF-1nF¯(x+μλ(t)-nμ)P(N(t)=n)≤(1+ϵ)LF-1F¯(x+μλ(t)-μ(1-δ)λ(t))×∑|n-λ(t)|≤δλ(t)‍nP(N(t)=n)≤(1+ϵ)LF-1F¯((1+δμγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t)).
On the other hand, for x≥γλ(t)≥(γ/(1+δ))n,
(56)J2≥(1-ϵ)LF∑|n-λ(t)|≤δλ(t)‍nF¯((1-μδγ)x)P(N(t)=n)=(1-ϵ)LFF¯((1-μδγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t)).
Similarly, both (50) and (52) hold in this case.Lemma 14.
Assume that AssumptionsN1 and N2 hold, and then for any γ>0,
(57)J3:=∑n>(1+δ)λ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)=o(λ(t)F¯(x))
holds uniformly for x>γλ(t).Proof.
We prove this lemma by splitting into three cases like the former lemmas’ proof.
(i)μ≥0 and γ>0.
Forp>𝕁F≥1, by Lemma 8, there exists a constant C>0 such that
(58)J3=∑n>(1+δ)λ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)≤∑n>(1+δ)λ(t)‍P(Sn>x)P(N(t)=n)≤∑n>(1+δ)λ(t)‍(nF¯(xp)+C(nx)p)P(N(t)=n)=F¯(xp)EN(t)I(N(t)>(1+δ)λ(t))+Cx-pEN(t)pI(N(t)>(1+δ)λ(t))≍F¯(x)EN(t)I(N(t)>(1+δ)λ(t))+x-pEN(t)pI(N(t)>(1+δ)λ(t))=o(λ(t)F¯(x)),
where in the last step we have used Lemma 2, Assumption N1 and relation (38).
(ii)μ<0 and γ>|μ|.
Similarly, by Lemma8, there exists a constant C>0 such that
(59)J3≤∑n>(1+δ)λ(t)‍P(Sn>(1+μγ)x)P(N(t)=n)≤∑n>(1+δ)λ(t)‍(nF¯((1+μ/γ)xp)≤∑n>(1+δ)λ(t)‍+C(n(1+μ/γ)x)p)P(N(t)=n)=F¯((1+μ/γ)xp)EN(t)I(N(t)>(1+δ)λ(t))+C((1+μγ)x)-pEN(t)pI(N(t)>(1+δ)λ(t))≍F¯(x)EN(t)I(N(t)>(1+δ)λ(t))+x-pEN(t)pI(N(t)>(1+δ)λ(t))=o(λ(t)F¯(x)).
(iii)μ<0 and γ∈(0,|μ|].
Note thatx+μλ(t)-nμ≥(γ+μ)λ(t)-μn≥((γ+μ)/(1+δ)-μ)n. Therefore, Lemma 6 implies that
(60)J3≤∑n>(1+ϵ)λ(t)‍(1+δ)LF-1nF¯(x+μλ(t)-nμ)P(N(t)=n)≤(1+ϵ)LF-1F¯(x+μλ(t)-μ(1+ϵ)λ(t))EN(t)I×(N(t)>(1+ϵ)λ(t))=o(λ(t)F¯(x)).


## 3.1. Main Theorem

Theorem 9.
Let{Xk,k≥1} be a sequence of real-valued END random variables with common distribution F∈ℒ∩𝒟 and finite mean μ, satisfying (18). Let {N(t),t≥0} be a nonnegative integer-valued process independent of {Xk,k≥1}, and assume that N(t) satisfies
(31)AssumptionN1:EN(t)pI(N(t)>(1+δ)λ(t))=O(λ(t))
for some p>𝕁F and
(32)AssumptionN2:P(N(t)≤(1-δ)λ(t))=o(λ(t)F¯(λ(t)))
for all 0<δ<1. SN(t) is defined by (1), and then for any γ>0,
(33)LF2λ(t)F¯(x)≲P(SN(t)-μλ(t)>x)≲LF-2λ(t)F¯(x)
holds uniformly for x≥γλ(t).Remark 10.
From the following proof, it is easy to see that (33) holds uniformly for x≥γλ(t) and γ>|μ| if N(t) just satisfies Assumption N1.Corollary 11.
Under the assumptions of Theorem9 and if F∈𝒞, then for any γ>0,
(34)P(SN(t)-μλ(t)>x)~λ(t)F¯(x)
holds uniformly for x≥γλ(t).From Remark10, if μ>0, Corollary 11 is just the same as Theorem 3.1 of Chen et al. [9]. The previous results extend Chen et al. [9] and Chen and Zhang [7].


## 3.2. Proof of Theorem9

In the sequel,C always represents an absolute positive constant, which may vary from different places.In order to prove Theorem9, we divide the random sum into three parts as follows:
(35)P(SN(t)-μλ(t)>x)=∑n=1∞‍P(Sn-μλ(t)>x)P(N(t)=n)=(∑n<(1-δ)λ(t)‍+∑|n-λ(t)|≤δλ(t)‍+∑n>(1+δ)λ(t)‍)×P(Sn-μλ(t)>x)P(N(t)=n),
where δ∈(0,1) is an arbitrarily fixed number to be specified later.By Tang et al. [3], Assumption N1 implies that
(36)N(t)λ(t)→p1;
then by the dominated convergence theorem,
(37)limt→∞E[N(t)λ(t)I(|N(t)-λ(t)|≤δλ(t))]=1.
Therefore, for any 0<δ<1,
(38)E[N(t)I(N(t)>(1+δ)λ(t))]=o(λ(t)),(39)E[N(t)I(N(t)<(1-δ)λ(t))]=o(λ(t)).Therefore, Theorem9 originates from the following three lemmas.Lemma 12.
Assume that AssumptionsN1 and N2 hold; then for any γ>0,
(40)J1:=∑n<(1-δ)λ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)=o(λ(t)F¯(x))
holds uniformly for x≥γλ(t).Proof.
In order to prove this lemma, we consider the following three cases.
(i)μ≥0 and γ>0.
Choose0<δ<1 such that δμ<γ. Note that n<(1-δ)λ(t), x≥γλ(t)>(γ/(1-δ))n. By Lemma 7,
(41)J1=∑n<(1-δ)λ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)=∑n<(1-δ)λ(t)‍P(Sn-nμ>x+μ(λ(t)-n))P(N(t)=n)≤∑n<(1-δ)λ(t)‍P(Sn-nμ>x)P(N(t)=n)≤CF¯(x)∑n<(1-δ)λ(t)‍nP(N(t)=n)=CF¯(x)EN(t)I(N(t)<(1-δ)λ(t))=o(λ(t)F¯(x)),
and the last equality holds by relation (39).
(ii)μ<0 and γ>|μ|.
Chooseδ such that (γ+μ)/(1-δ)-μ>0. Note that x+μλ(t)-nμ≥((γ+μ)/(1-δ)-μ)n. Mimicking the proof of (i) and by Lemma 7, we have
(42)J1≤∑n<(1-δ)λ(t)‍CnF¯(x+μλ(t)-nμ)P(N(t)=n)≤CF¯(x+μλ(t))EN(t)I(N(t)<(1-δ)λ(t))=o(λ(t)F¯(x)),
where in the last step, we used the relation F¯(x+μλ(t))≤F¯((1+μ/γ)x)≍F¯(x).
(iii)μ<0 and γ∈(0,|μ|].
Letγ1>|μ|. For x≥γ1λ(t), we have x+μλ(t)-nμ>|μ|n. By Lemma 7,
(43)J1≤∑n<(1-δ)λ(t)‍CnF¯(x+μλ(t)-nμ)P(N(t)=n)≤CF¯(x+μλ(t))∑n<(1-δ)λ(t)‍nP(N(t)=n)=CF¯(x+μλ(t))EN(t)I(N(t)<(1-δ)λ(t))=o(λ(t)F¯(x)),
where the last step can be verified as F¯(x+μλ(t))≤F¯((1+μ/γ1)x)≍F¯(x).
Forγλ(t)≤x<γ1λ(t), note that F¯(x)≥F¯(γ1λ(t))≍F¯(λ(t)). Hence, by Assumption N2, the inequality
(44)J1≤P(N(t)≤(1-δ)λ(t))=o(1)λ(t)F¯(λ(t))=o(λ(t)F¯(x))
still holds uniformly for all γλ(t)≤x<γ1λ(t). As a result, the relation
(45)J1=o(λ(t)F¯(x))
holds uniformly for all x≥γλ(t).Lemma 13.
Assume that AssumptionsN1 and N2 hold; then for any γ>0,
(46)LF2λ(t)F¯(x)≲J2:=∑|n-λ(t)|≤δλ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)≲LF-2λ(t)F¯(x)
holds uniformly for x≥γλ(t).Proof.
Similar to the proof of Lemma12, we split into three cases.
(i)μ≥0 and γ>0.
Firstly, for|n-λ(t)|≤δλ(t), then λ(t)-n≥-δλ(t). For x≥γλ(t),
(47)J2=∑|n-λ(t)|≤δλ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)=∑|n-λ(t)|≤δλ(t)‍P(Sn-nμ>x+μ(λ(t)-n))P(N(t)=n)≤∑|n-λ(t)|≤δλ(t)‍P(Sn-nμ>x-μδλ(t))P(N(t)=n)≤∑|n-λ(t)|≤δλ(t)‍P(Sn-nμ>(1-μδγ)x)P(N(t)=n).
By Lemma 6, for any ϵ>0 and sufficient large t,x≥γλ(t)≥(γ/(1+δ))n,
(48)P(Sn-nμ>(1-μδγ)x)≤(1+ϵ)LF-1nF¯((1-μδγ)x).
Then,
(49)J2≤(1+ϵ)LF-1F¯((1-μδγ)x)∑|n-λ(t)|≤δλ(t)‍nP(N(t)=n)=(1+ϵ)LF-1F¯((1-μδγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t));
thus by F∈ℒ∩𝒟, letting δ→0, we have limδ→0F¯((1-μδ/γ)x)/F¯(x)=LF-1. Combining this and (37),
(50)limsupt→∞supx≥γλ(t)J2λ(t)F¯(x)≤LF-2.
On the other hand, by the similar way, for |n-λ(t)|≤δλ(t), then λ(t)-n≤δλ(t),
(51)J2=∑|n-λ(t)|≤δλ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)≥∑|n-λ(t)|≤δλ(t)‍P(Sn-nμ>x+μδλ(t))P(N(t)=n)≥∑|n-λ(t)|≤δλ(t)‍P(Sn-nμ>(1+μδγ)x)P(N(t)=n)≥(1-ϵ)LFγF(μ)F¯((1+δμγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t)).
We have
(52)liminft→∞infx>γλ(t)J2λ(t)F¯(x)≥LF2γF(μ).
Combining (50) and (52), we complete the proof of the lemma in this case.
(ii)μ<0 and γ>|μ|.
Similarly, for anyϵ>0 and sufficient t,x≥γλ(t),
(53)J2≤∑|n-λ(t)|≤δλ(t)‍(1+ϵ)LF-1nF¯(x+μλ(t)-nμ)P(N(t)=n)≤(1+ϵ)LF-1F¯(x+μλ(t)-μ(1-δ)λ(t))×∑|n-λ(t)|≤δλ(t)‍nP(N(t)=n)≤(1+ϵ)LF-1γF-2(-μ)F¯((1+δμγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t)),J2≥(1-ϵ)LFγF(-μ)F¯((1-δμγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t)).
Analogously, we can obtain
(54)limsupt→∞supx≥γλ(t)J2λ(t)F¯(x)≤LF-2γF-2(-μ),liminft→∞infx>γλ(t)J2λ(t)F¯(x)≥LF2γF(-μ).
(iii)μ<0 and γ∈(0,|μ|].
Sinceγ∈(0,|μ|], note that x+μλ(t)-nμ≥(γ+μ)λ(t)-μn≥((γ+μ)/(1-δ)-μ)n, and it follows from Lemma 6 that for all x≥γλ(t),
(55)J2≤∑|n-λ(t)|≤δλ(t)‍(1+ϵ)LF-1nF¯(x+μλ(t)-nμ)P(N(t)=n)≤(1+ϵ)LF-1F¯(x+μλ(t)-μ(1-δ)λ(t))×∑|n-λ(t)|≤δλ(t)‍nP(N(t)=n)≤(1+ϵ)LF-1F¯((1+δμγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t)).
On the other hand, for x≥γλ(t)≥(γ/(1+δ))n,
(56)J2≥(1-ϵ)LF∑|n-λ(t)|≤δλ(t)‍nF¯((1-μδγ)x)P(N(t)=n)=(1-ϵ)LFF¯((1-μδγ)x)EN(t)I×(|N(t)-λ(t)|≤δλ(t)).
Similarly, both (50) and (52) hold in this case.Lemma 14.
Assume that AssumptionsN1 and N2 hold, and then for any γ>0,
(57)J3:=∑n>(1+δ)λ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)=o(λ(t)F¯(x))
holds uniformly for x>γλ(t).Proof.
We prove this lemma by splitting into three cases like the former lemmas’ proof.
(i)μ≥0 and γ>0.
Forp>𝕁F≥1, by Lemma 8, there exists a constant C>0 such that
(58)J3=∑n>(1+δ)λ(t)‍P(Sn-μλ(t)>x)P(N(t)=n)≤∑n>(1+δ)λ(t)‍P(Sn>x)P(N(t)=n)≤∑n>(1+δ)λ(t)‍(nF¯(xp)+C(nx)p)P(N(t)=n)=F¯(xp)EN(t)I(N(t)>(1+δ)λ(t))+Cx-pEN(t)pI(N(t)>(1+δ)λ(t))≍F¯(x)EN(t)I(N(t)>(1+δ)λ(t))+x-pEN(t)pI(N(t)>(1+δ)λ(t))=o(λ(t)F¯(x)),
where in the last step we have used Lemma 2, Assumption N1 and relation (38).
(ii)μ<0 and γ>|μ|.
Similarly, by Lemma8, there exists a constant C>0 such that
(59)J3≤∑n>(1+δ)λ(t)‍P(Sn>(1+μγ)x)P(N(t)=n)≤∑n>(1+δ)λ(t)‍(nF¯((1+μ/γ)xp)≤∑n>(1+δ)λ(t)‍+C(n(1+μ/γ)x)p)P(N(t)=n)=F¯((1+μ/γ)xp)EN(t)I(N(t)>(1+δ)λ(t))+C((1+μγ)x)-pEN(t)pI(N(t)>(1+δ)λ(t))≍F¯(x)EN(t)I(N(t)>(1+δ)λ(t))+x-pEN(t)pI(N(t)>(1+δ)λ(t))=o(λ(t)F¯(x)).
(iii)μ<0 and γ∈(0,|μ|].
Note thatx+μλ(t)-nμ≥(γ+μ)λ(t)-μn≥((γ+μ)/(1+δ)-μ)n. Therefore, Lemma 6 implies that
(60)J3≤∑n>(1+ϵ)λ(t)‍(1+δ)LF-1nF¯(x+μλ(t)-nμ)P(N(t)=n)≤(1+ϵ)LF-1F¯(x+μλ(t)-μ(1+ϵ)λ(t))EN(t)I×(N(t)>(1+ϵ)λ(t))=o(λ(t)F¯(x)).


## 4. Applications for Random Sums for a Dependent Compound Renewal Risk Model

### 4.1. The Dependent Compound Renewal Risk Model

From a realistic point of view, we further generalize such work to a much more realistic model including premium income process. The premium income process depends not only on the number of customers who buy the insurance portfolios but also on the premium size process. The risk model has the following structure.(1)
The individual claim sizes{Xk,k≥1} are END random variables with common distribution F and finite mean μ=EX1.(2)
The number of claims in the interval[0,t] is denoted by {N(t),t≥0}. Suppose that {N(t),t≥0} is a nonnegative integer-valued process with mean function λ(t) which tends to ∞ as t→∞ and independent of {Xk,k≥1}.(3)
The number of customers who buy the insurance portfolios within the time interval(0,t] is denoted by {M(t),t≥0}. Assume that {M(t),t≥0} is a strictly stationary renewal counting process with mean function EM(t)=βt,β>0.(4)
The premium size process{Yk,k≥1} is a sequence of nonnegative UND random variables.Suppose thatu>0 is the initial reserve of an insurance company. Thus, the risk reserve process {R(t),t≥0} is given by
(61)R(t)=u+∑k=1M(t)‍Yk-∑k=1N(t)‍Xk,t≥0,
while the claim surplus process {S(t),t≥0} is
(62)S(t)=∑k=1N(t)‍Xk-∑k=1M(t)‍Yk,t≥0.
In addition, the random sequences {M(t),t≥0},{Yk,k≥1},{Xk,k≥1}, and {N(t),t≥0} are mutually independent. This model (62) is called generalized dependent compound renewal risk model. For this model (62), the net profit condition is that μλ(t)<βtEY1.Theorem 15.
For the generalized dependent compound renewal risk model (62), let {Xk,k≥1} be a sequence of END random variables with common distribution F∈ℒ∩𝒟 and finite mean μ, satisfying (18), and let {Yk,k≥1} be a sequence of UND nonnegative random variables. Assume that N(t) satisfies Assumptions N1 and N2, and then for any γ>EY1,
(63)LF3λ(t)F¯(x)≲P(S(t)-ES(t)>x)≲LF-3λ(t)F¯(x)
holds uniformly for x≥γλ(t)∨γβt.Applying Theorem9 to the generalized dependent compound renewal risk model (62), we can obtain the previous theorem by the following lemmas.By the same argument as in Theorem15, we can easily obtain the following corollary which extends Hu’s result [16].Corollary 16.
Under the conditions of Theorem15, further assume that N(t) is a Poisson process with intensity λ>0, and then for any γ>0 with γλ>βEY1,
(64)LF3λtF¯(x)≲P(S(t)-ES(t)>x)≲LF-3λtF¯(x)
holds uniformly for x≥γλt.Remark 17.
If we assume thatF∈𝒞 in Theorem 15, we have that
(65)P(S(t)-ES(t)>x)~λ(t)F¯(x)
holds uniformly for x≥γλ(t)∨γβt.From the previous two theorems, we find out that the asymptotic behavior of precise large deviations of random sums is insensitive to the extended negative dependence.


### 4.2. Proof of Theorem15

Observing that{M(t),t≥0} is a renewal counting process, EM(t)=βt. According to the strong law of large numbers for UND random variables (see Matuła [17, Theorem 1]), when t→∞, we have
(66)1t∑j=1M(t)‍Yj=M(t)t·1M(t)∑j=1M(t)‍Yj→βEY1.
Consequently, there is a positive function ϵ(t) such that ϵ(t)→0, and
(67)P(|∑j=1M(t)‍Yj-βtEY1|>ϵ(t)βtEY1)=o(1).
For convenience, we denote that J(x;t)=J(x)=P(∑j=1M(t)‍Yj≤x). For the model (62), we have
(68)(S(t)-ES(t)>x)=P(SN(t)-∑j=1M(t)‍Yj-μλ(t)+EM(t)EY1>x)=∫0∞‍P(SN(t)-μλ(t)>x-βtEY1+y)dJ(y).
In order to prove the theorem, we divide this integral into three parts similarly as follows:
(69)∫0∞‍=∫y-βtEY1<-ϵ(t)βtEY1‍+∫|y-βtEY1|≤ϵ(t)βtEY1‍+∫y-βtEY1>ϵ(t)βtEY1‍.We proceed a series of lemmas to prove Theorem15.Lemma 18.
Forϵ(t) in (67), one has for γ>EY1 that
(70)I1=∫y-βtEY1<-ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x=∫y-βtEY1<-ϵ(t)βtEY1‍P-βtEY1+y)dJ(y)=o(λ(t)F¯(x))
holds uniformly for x≥γλ(t)∨γβt.Proof.
Forγ>EY1, then 1-EY1/γ>0. For y≥0, x≥γλ(t)∨γβt, then t≤x/γβ, and applying Theorem 9, we have
(71)I1=∫y-βtEY1<-ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x-βtEY1=∫y-βtEY1<-ϵ(t)βtEY1‍P+ySN(t)-μλ(t))dJ(y)≤∫y-βtEY1<-ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x≤∫y-βtEY1<-ϵ(t)βtEY1‍P-βtEY1SN(t)-μλ(t))dJ(y)≤∫y-βtEY1<-ϵ(t)βtEY1‍P((1-EY1γ)SN(t)-μλ(t)≤∫y-βtEY1<-ϵ(t)βtEY1‍P>x(1-EY1γ))dJ(y)≲LF-2λ(t)F¯(x(1-EY1γ))×P(∑i=1M(t)‍Yi-βtEY1<-ϵ(t)βtEY1)≤Cλ(t)F¯(x)o(1)=o(λ(t)F¯(x)),
where C>0 is a constant, the last second step holds by F∈ℒ∩𝒟 and relation (67).Lemma 19.
Forϵ(t) in (67), one has for γ>EY1 that
(72)LF3λ(t)F¯(x)≲I2=∫|y-βtEY1|≤ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x≲I2=∫|y-βtEY1|≤ϵ(t)βtEY1‍P-βtEY1+ySN(t)-μλ(t))dJ(y)≲LF-3λ(t)F¯(x)
holds uniformly for x≥γλ(t)∨γβt.Proof.
For anyx≥γλ(t)∨γβt and |y-βtEY1|≤ϵ(t)βtEY1, noting that ϵ(t)→0 as t→∞, we have for sufficient large t,
(73)x-βtEY1+y≥x-ϵ(t)βtEY1≥(1-ϵ(t)EY1γ)x>(1-EY1γ)x.
Again using Theorem9, we conclude that
(74)LF2≲P(SN(t)-μλ(t)>x-βtEY1+y)λ(t)F¯(x-βtEY1+y)≲LF-2
holds uniformly for x≥γλ(t). Moreover, |y-βtEY1|≤ϵ(t)βtEY1, and then |y-βtEY1|=o(λ(t)) as t→∞; by Lemma 3, we have that
(75)LFF¯(x)≲F¯(x-βtEY1+y)≲LF-1F¯(x)
holds uniformly for x≥γλ(t).
Hence, combining (67), (74), and (75), we see that
(76)I2=∫|y-βtEY1|≤ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x-βtEY1=∫|y-βtEY1|≤ϵ(t)βtEY1‍P+ySN(t)-μλ(t))dJ(y)≲LF-2λ(t)F¯(x)×∫|y-βtEY1|≤ϵ(t)βtEY1‍F¯(x-βtEY1+y)F¯(x)dJ(y)≲LF-3λ(t)F¯(x)P(|∑i=1M(t)‍Yi-βtEY1|≤ϵ(t)βtEY1)~LF-3λ(t)F¯(x);
symmetrically,
(77)I2≳LF3λ(t)F¯(x).
Then the proof of Lemma 19 is finished.Lemma 20.
Forϵ(t) in(67), one has for γ>EY1 that
(78)I3=∫y-βtEY1>ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x-βtEY1=∫y-βtEY1>ϵ(t)βtEY1‍P+ySN(t)-μλ(t))dJ(y)=o(λ(t)F¯(x))
holds uniformly for x≥γλ(t)∨γβt.Proof.
Forx≥γλ(t)∨γβt, by Theorem 9, we see that
(79)I3=∫y-βtEY1>ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x-βtEY1=∫y-βtEY1>ϵ(t)βtEY1‍P+ySN(t))dJ(y)≤∫y-βtEY1>ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x)dJ(y)≲LF-2λ(t)F¯(x)P(∑i=1M(t)‍Yi-βtEY1>ϵ(t)βtEY1)=o(λ(t)F¯(x));
the last equality is from (67).


## 4.1. The Dependent Compound Renewal Risk Model

From a realistic point of view, we further generalize such work to a much more realistic model including premium income process. The premium income process depends not only on the number of customers who buy the insurance portfolios but also on the premium size process. The risk model has the following structure.(1)
The individual claim sizes{Xk,k≥1} are END random variables with common distribution F and finite mean μ=EX1.(2)
The number of claims in the interval[0,t] is denoted by {N(t),t≥0}. Suppose that {N(t),t≥0} is a nonnegative integer-valued process with mean function λ(t) which tends to ∞ as t→∞ and independent of {Xk,k≥1}.(3)
The number of customers who buy the insurance portfolios within the time interval(0,t] is denoted by {M(t),t≥0}. Assume that {M(t),t≥0} is a strictly stationary renewal counting process with mean function EM(t)=βt,β>0.(4)
The premium size process{Yk,k≥1} is a sequence of nonnegative UND random variables.Suppose thatu>0 is the initial reserve of an insurance company. Thus, the risk reserve process {R(t),t≥0} is given by
(61)R(t)=u+∑k=1M(t)‍Yk-∑k=1N(t)‍Xk,t≥0,
while the claim surplus process {S(t),t≥0} is
(62)S(t)=∑k=1N(t)‍Xk-∑k=1M(t)‍Yk,t≥0.
In addition, the random sequences {M(t),t≥0},{Yk,k≥1},{Xk,k≥1}, and {N(t),t≥0} are mutually independent. This model (62) is called generalized dependent compound renewal risk model. For this model (62), the net profit condition is that μλ(t)<βtEY1.Theorem 15.
For the generalized dependent compound renewal risk model (62), let {Xk,k≥1} be a sequence of END random variables with common distribution F∈ℒ∩𝒟 and finite mean μ, satisfying (18), and let {Yk,k≥1} be a sequence of UND nonnegative random variables. Assume that N(t) satisfies Assumptions N1 and N2, and then for any γ>EY1,
(63)LF3λ(t)F¯(x)≲P(S(t)-ES(t)>x)≲LF-3λ(t)F¯(x)
holds uniformly for x≥γλ(t)∨γβt.Applying Theorem9 to the generalized dependent compound renewal risk model (62), we can obtain the previous theorem by the following lemmas.By the same argument as in Theorem15, we can easily obtain the following corollary which extends Hu’s result [16].Corollary 16.
Under the conditions of Theorem15, further assume that N(t) is a Poisson process with intensity λ>0, and then for any γ>0 with γλ>βEY1,
(64)LF3λtF¯(x)≲P(S(t)-ES(t)>x)≲LF-3λtF¯(x)
holds uniformly for x≥γλt.Remark 17.
If we assume thatF∈𝒞 in Theorem 15, we have that
(65)P(S(t)-ES(t)>x)~λ(t)F¯(x)
holds uniformly for x≥γλ(t)∨γβt.From the previous two theorems, we find out that the asymptotic behavior of precise large deviations of random sums is insensitive to the extended negative dependence.


## 4.2. Proof of Theorem15

Observing that{M(t),t≥0} is a renewal counting process, EM(t)=βt. According to the strong law of large numbers for UND random variables (see Matuła [17, Theorem 1]), when t→∞, we have
(66)1t∑j=1M(t)‍Yj=M(t)t·1M(t)∑j=1M(t)‍Yj→βEY1.
Consequently, there is a positive function ϵ(t) such that ϵ(t)→0, and
(67)P(|∑j=1M(t)‍Yj-βtEY1|>ϵ(t)βtEY1)=o(1).
For convenience, we denote that J(x;t)=J(x)=P(∑j=1M(t)‍Yj≤x). For the model (62), we have
(68)(S(t)-ES(t)>x)=P(SN(t)-∑j=1M(t)‍Yj-μλ(t)+EM(t)EY1>x)=∫0∞‍P(SN(t)-μλ(t)>x-βtEY1+y)dJ(y).
In order to prove the theorem, we divide this integral into three parts similarly as follows:
(69)∫0∞‍=∫y-βtEY1<-ϵ(t)βtEY1‍+∫|y-βtEY1|≤ϵ(t)βtEY1‍+∫y-βtEY1>ϵ(t)βtEY1‍.We proceed a series of lemmas to prove Theorem15.Lemma 18.
Forϵ(t) in (67), one has for γ>EY1 that
(70)I1=∫y-βtEY1<-ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x=∫y-βtEY1<-ϵ(t)βtEY1‍P-βtEY1+y)dJ(y)=o(λ(t)F¯(x))
holds uniformly for x≥γλ(t)∨γβt.Proof.
Forγ>EY1, then 1-EY1/γ>0. For y≥0, x≥γλ(t)∨γβt, then t≤x/γβ, and applying Theorem 9, we have
(71)I1=∫y-βtEY1<-ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x-βtEY1=∫y-βtEY1<-ϵ(t)βtEY1‍P+ySN(t)-μλ(t))dJ(y)≤∫y-βtEY1<-ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x≤∫y-βtEY1<-ϵ(t)βtEY1‍P-βtEY1SN(t)-μλ(t))dJ(y)≤∫y-βtEY1<-ϵ(t)βtEY1‍P((1-EY1γ)SN(t)-μλ(t)≤∫y-βtEY1<-ϵ(t)βtEY1‍P>x(1-EY1γ))dJ(y)≲LF-2λ(t)F¯(x(1-EY1γ))×P(∑i=1M(t)‍Yi-βtEY1<-ϵ(t)βtEY1)≤Cλ(t)F¯(x)o(1)=o(λ(t)F¯(x)),
where C>0 is a constant, the last second step holds by F∈ℒ∩𝒟 and relation (67).Lemma 19.
Forϵ(t) in (67), one has for γ>EY1 that
(72)LF3λ(t)F¯(x)≲I2=∫|y-βtEY1|≤ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x≲I2=∫|y-βtEY1|≤ϵ(t)βtEY1‍P-βtEY1+ySN(t)-μλ(t))dJ(y)≲LF-3λ(t)F¯(x)
holds uniformly for x≥γλ(t)∨γβt.Proof.
For anyx≥γλ(t)∨γβt and |y-βtEY1|≤ϵ(t)βtEY1, noting that ϵ(t)→0 as t→∞, we have for sufficient large t,
(73)x-βtEY1+y≥x-ϵ(t)βtEY1≥(1-ϵ(t)EY1γ)x>(1-EY1γ)x.
Again using Theorem9, we conclude that
(74)LF2≲P(SN(t)-μλ(t)>x-βtEY1+y)λ(t)F¯(x-βtEY1+y)≲LF-2
holds uniformly for x≥γλ(t). Moreover, |y-βtEY1|≤ϵ(t)βtEY1, and then |y-βtEY1|=o(λ(t)) as t→∞; by Lemma 3, we have that
(75)LFF¯(x)≲F¯(x-βtEY1+y)≲LF-1F¯(x)
holds uniformly for x≥γλ(t).
Hence, combining (67), (74), and (75), we see that
(76)I2=∫|y-βtEY1|≤ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x-βtEY1=∫|y-βtEY1|≤ϵ(t)βtEY1‍P+ySN(t)-μλ(t))dJ(y)≲LF-2λ(t)F¯(x)×∫|y-βtEY1|≤ϵ(t)βtEY1‍F¯(x-βtEY1+y)F¯(x)dJ(y)≲LF-3λ(t)F¯(x)P(|∑i=1M(t)‍Yi-βtEY1|≤ϵ(t)βtEY1)~LF-3λ(t)F¯(x);
symmetrically,
(77)I2≳LF3λ(t)F¯(x).
Then the proof of Lemma 19 is finished.Lemma 20.
Forϵ(t) in(67), one has for γ>EY1 that
(78)I3=∫y-βtEY1>ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x-βtEY1=∫y-βtEY1>ϵ(t)βtEY1‍P+ySN(t)-μλ(t))dJ(y)=o(λ(t)F¯(x))
holds uniformly for x≥γλ(t)∨γβt.Proof.
Forx≥γλ(t)∨γβt, by Theorem 9, we see that
(79)I3=∫y-βtEY1>ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x-βtEY1=∫y-βtEY1>ϵ(t)βtEY1‍P+ySN(t))dJ(y)≤∫y-βtEY1>ϵ(t)βtEY1‍P(SN(t)-μλ(t)>x)dJ(y)≲LF-2λ(t)F¯(x)P(∑i=1M(t)‍Yi-βtEY1>ϵ(t)βtEY1)=o(λ(t)F¯(x));
the last equality is from (67).



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*Source: 936301-2013-06-19.xml*