# A Note on the Tail Behavior of Randomly Weighted Sums with Convolution-Equivalently Distributed Random Variables

Authors: Yang Yang; Jun-feng Liu; Yu-lin Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the tailed asymptotic behavior of the randomly weighted sums with increments with convolution-equivalent distributions. Our obtained result can be directly applied to a discrete-time insurance risk model with insurance and financial risks and derive the asymptotics for the finite-time probability of the above risk model.


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## Body

## 1. Introduction and Main Result

Let{Xn,n≥1} be a sequence of independent and identically distributed (i.i.d.) real-valued random variables with common distribution F, and let {Yn,n≥1} be another sequence of i.i.d. nonnegative r.v.s with common distribution G and right endpoint x^G=sup{x:ℙ(Y1≤x)<1}. Assume that {Xn,n≥1} are independent of {Yn,n≥1}. In this paper, we are interested in the randomly weighted sum
(1)SnY=∑i=1n‍Xi∏j=1i‍Yj,n≥1.
This is because the study for the tail probability ℙ(SnY>x) can be directly applied to risk theory. Consider a discrete-time insurance risk model. Within period i, i≥1, the net insurance loss is denoted by a real-valued (r.v.) Xi. The insurer makes both risk-free and risky investments, leading to an overall stochastic  discounted factor Yi from time i to time i-1. In the terminology of Norberg [1], the sequences {Xn,n≥1} and {Yn,n≥1} are called the insurance and financial risks, respectively. Then, the randomly weighted sum SnY in (1) represents the stochastic  discounted value of aggregate net losses up to time n, n≥1. As usual, the probability of ruin by time n can be defined by
(2)Ψ(x,n)=ℙ(max1≤m≤n∑i=1m‍Xi∏j=1i‍Yj>x),n≥1,
where x≥0 is interpreted as the initial capital reserve of an insurance company. Clearly, for each n≥1,
(3)ℙ(SnY>x)≤Ψ(x,n)≤ℙ(∑i=1n‍Xi+∏j=1i‍Yj>x),
where Xi+=Xi1{Xi≥0} denotes the positive part of Xi, i≥1. If we can establish an asymptotic formula for ℙ(SnY>x) while doing so does not require F(0-)>0, then the same asymptotic formula should hold for the right-hand side of (3) as well. In this way the ruin probability Ψ(x,n) has the same asymptotic behavior as that of the tail probability ℙ(SnY>x) as x tends to infinity.There has been a vast amount of literature studying the asymptotic behavior of the tail probability of the randomly weighted sumSnY. Many works have considered the heavy-tailed case; that is, the distribution F of X belongs to some classes of heavy-tailed distributions, even under some dependence structures. For example, one can refer to Tang and Tsitsiashvili [2, 3], Wang and Tang [4], Zhang et al. [5], Shen et al. [6], Chen and Yuen [7], Gao and Wang [8], and Yi et al. [9] among others for some details in this direction, where the distribution F is heavily heavy tailed; as for some lightly heavy-tailed distribution F, some related results were obtained by Tang and Tsitsiashvili [3, 10], Chen and Su [11], Hashorva et al. [12], Yang et al. [13], Yang and Hashorva [14], and Yang and Wang [15] among others. We pointed out that Tang and Tsitsiashvili [3] achieved some interesting results on the asymptotics for the tail probability ℙ(SnY>x) in some cases where  F belongs to the intersection between the subexponential distribution class and the rapidly varying distribution class.In this paper, we aim to consider the light-tailed case, more exactly, to investigate the asymptotic behavior of the tail probability of the randomly weighted sums with increments with convolution-equivalent distributions.Hereafter, all the limit relationships hold forx tending to infinity. For two positive functions a(x) and b(x), we write a(x)~b(x) if lima(x)/b(x)=1; write a(x)=o(b(x)) if lima(x)/b(x)=0; and write a(x)=O(b(x)) if limsupa(x)/b(x)<∞.Firstly we introduce some definitions on some classes of convolution-equivalent distributions. A distributionV on [0,∞) belongs to the class of convolution-equivalent distributions, denoted by 𝒮(γ), γ≥0, if for any y∈ℝ,
(4)limV¯(x-y)V¯(x)=eγy,(5)limV*2¯(x)V¯(x)=2∫0∞‍eγuV(du)<∞,
where V*2 denotes the convolution of V with itself. More generally, a distribution V on ℝ belongs to the class 𝒮(γ), γ≥0, if and only if its right-hand distribution V+(x)=V(x)1{x≥0} belongs to this class; see Corollary 2.1 of Pakes [16]. The class 𝒮:=𝒮(0) is called the class of subexponential distributions. A distribution V on ℝ belongs to the class ℒ(γ), γ≥0 if only relation (4) holds. In the case γ=0, we say that ℒ:=ℒ(0) is the class of long-tailed distributions. Similarly, a positive function f(·) is said to be long tailed if limf(x-y)/f(x)=1 for any y∈ℝ. Clearly, if a distribution V∈ℒ, then its tail probability V¯(x) is long tailed. Closely related is the class 𝒜, which was introduced by Konstantinides et al. [17]. A distribution V on ℝ belongs to the class 𝒜 if V is subexponential, and, for some y>1,
(6)limsupV¯(xy)V¯(x)<1.
Clearly, all distributions in the classes 𝒜, 𝒮, and ℒ are heavy tailed. A distribution V of r.v. ξ is said to be heavy tailed if 𝔼esξ=∞ for any s>0; otherwise it is said to be light tailed.For eachn≥1, denote the distribution of Xn∏j=1n‍Yj by Hn, by convention, H=H1. Now we state our main result as follows.Theorem 1.
IfF∈𝒮(γ) for some γ>0, x^G=∞, and, for all u>0,
(7)G¯(ux)=o(H¯(x)),
then, for each n≥1,
(8)ℙ(SnY>x)~Hn¯(x).Remark 2.
We remark that Tang [18] considered a similar result for 0<x^G<∞, whereas Theorem 1 deals with the case with x^G=∞ for a complement.Remark 3.
In Theorem1, relation (7) is a mild condition. According to Corollary 1.1 of Tang [19], relation (7) can be further implied by either(a)
G¯(vx)=o(G¯(x)) for some v>1 or(b)
G¯(vx)=o(F¯(x)) for some v>0.


## 2. Proof of the Main Result

We start this section by a series of lemmas. The first two lemmas are due to Lemma 3.2 and Theorem 2.1 of Tang [20].Lemma 4.
For two distributionsG and H with G¯(x)>0 and H¯(x)>0 for all x≥0, relation (7) holds for each u>0, if and only if there is a nonnegative function a(·) such that
(9)a(x)↗∞,a(x)x↘0,G¯(a(x))=o(H¯(x)).Lemma 5.
Consider the productXY. The distribution H of the product belongs to the class 𝒜 if and only if F∈𝒜 and relation (7) holds for all u>0.Tang [19] obtained an interesting result to show that a light-tailed random variable can be transferred into a heavy-tailed one through multiplier.Lemma 6.
Consider the productXY with F∈𝒮(γ) for some γ>0 and x^G=∞. If relation (7) holds for all u>0, then H∈𝒮.The last lemma can be found in, for example, Theorem 3.14 of Foss et al. [21].Lemma 7.
Let a reference distributionV on ℝ belong to the class 𝒮. Assume that distributions V1,…,Vn on ℝ  satisfy that, for each i=1,…,n, the function V¯+Vi¯ is long tailed and Vi¯(x)=O(V¯(x)). Then, it holds that
(10)V1*⋯*Vn¯(x)=V1¯(x)+⋯+Vn¯(x)+o(V¯(x)).Proof of Theorem1.
Now we begin to prove the main result of Theorem1.
For eachn≥1, write
(11)Tn:=∑i=1n‍Xi∏j=in‍Yj=dSnY,
where =d stands for equality in distribution. Since F∈𝒮(γ)⊂ℒ(γ), γ>0, its tail distribution F¯ is rapidly varying in the sense that
(12)limF¯(xy)F¯(x)=0,∀y>1.
By Lemma 6, we get that H∈𝒮. Further, by Lemma 4, there exists a nonnegative function a(·) such that (9) holds. Thus, by (9) and (12), for any y>1,
(13)limsupH+¯(xy)H+¯(x)=limsup(∫0a(x)‍+∫a(x)∞‍)F¯(xy/u)G(du)H+¯(x)≤limsup∫0a(x)‍F¯(xy/u)G(du)∫0a(x)‍F¯(x/u)G(du)+limsupG¯(a(x))H+¯(x)≤limsupsup0≤u≤a(x)F¯(xy/u)F¯(x/u)=limsupF¯(xy)F¯(x)=0<1,
which, together with H∈𝒮, implies that H∈𝒜.
We proceed to prove relation (8) by induction on n. Trivially, the distribution H1=H of T1 or S1 belongs to the class 𝒜, and relation (8) holds for n=1. Assume that Hn∈𝒜 and (8) holds for n. We aim to prove that Hn+1∈𝒜 and (8) holds for n+1, which, by (11), is equivalent to
(14)ℙ(Tn+1>x)~Hn+1¯(x).
First of all, according to Lemma 2.17 of Foss et al. [21] and Hn∈𝒜⊂ℒ, we have, that for any s>0,
(15)limHn¯(x)esx=∞,
which, together with F∈𝒮(γ),γ>0, implies that
(16)F¯(x)=o(Hn¯(x)).
By (16) and Hn∈𝒜⊂ℒ, we have that for any y∈ℝ,
(17)limHn¯(x-y)+F¯(x-y)Hn¯(x)+F¯(x)=lim((Hn¯(x-y)Hn¯(x))+(F¯(x-y)Hn¯(x-y))·(Hn¯(x-y)Hn¯(x)))×(1+(F¯(x)Hn¯(x)))-1=1,
which shows that the function Hn¯(x)+F¯(x) is long tailed. Since Tn and Xn+1 are independent of each other, thus, by Hn∈𝒜⊂𝒮, we can apply Lemma 7 to  derive from the induction assumption and (16) that
(18)ℙ(Tn+Xn+1>x)=Hn¯(x)+F¯(x)+o(Hn¯(x))=(1+o(1))Hn¯(x).
For the above-mentioned nonnegative function a(·), from (9) and (18), we obtain that
(19)ℙ(Tn+1>x)=ℙ((Tn+Xn+1)Yn+1>x)=(∫0a(x)‍+∫a(x)∞‍)ℙ(Tn+Xn+1>xu)G(du)=(1+o(1))∫0a(x)‍Hn¯(xu)G(du)+o(H¯(x))=(1+o(1))Hn+1¯(x),
where the last step used the fact that H¯(x)=O(Hn+1¯(x)), because, for any x≥0,
(20)Hn+1¯(x)≥ℙ(Xn+1∏j=1n+1‍Yj>x,Y1>1,…,Yn>1)≥H¯(x)(G¯(1))n,
and G¯(1)>0 by x^G=∞. Relation (19) means that (14) holds.
Finally, by (9) and (20), we have that
(21)G¯(a(x))=o(Hn+1¯(x)),
from which and Hn∈𝒜, Lemma 5 gives that Hn+1∈𝒜.
This completes the proof of Theorem1.



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*Source: 273217-2013-12-05.xml*