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

and Applied Analysis 3 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 lim F (xy) F (x) = 0, ∀y > 1. (12) 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, lim sup H (xy) H (x)


Introduction and Main Result
Let {  ,  ≥ 1} be a sequence of independent and identically distributed (i.i.d.) real-valued random variables with common distribution , and let {  ,  ≥ 1} be another sequence of i.i.d.nonnegative r.v.s with common distribution  and right endpoint x = sup{ : P( 1 ≤ ) < 1}.Assume that {  ,  ≥ 1} are independent of {  ,  ≥ 1}.In this paper, we are interested in the randomly weighted sum This is because the study for the tail probability P(   > ) can be directly applied to risk theory.Consider a discretetime insurance risk model.Within period ,  ≥ 1, the net insurance loss is denoted by a real-valued (r.v.)   .The insurer makes both risk-free and risky investments, leading to an overall stochastic discounted factor   from time  to time −1.
In the terminology of Norberg [1], the sequences {  ,  ≥ 1} and {  ,  ≥ 1} are called the insurance and financial risks, respectively.Then, the randomly weighted sum    in (1) represents the stochastic discounted value of aggregate net losses up to time ,  ≥ 1.As usual, the probability of ruin by time  can be defined by where  ≥ 0 is interpreted as the initial capital reserve of an insurance company.Clearly, for each  ≥ 1, where  +  =   1 {  ≥0} denotes the positive part of   ,  ≥ 1.If we can establish an asymptotic formula for P(   > ) while doing so does not require (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(, ) has the same asymptotic behavior as that of the tail probability P(   > ) as  tends to infinity.
There has been a vast amount of literature studying the asymptotic behavior of the tail probability of the randomly weighted sum    .Many works have considered the heavytailed case; that is, the distribution  of  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  is heavily heavy tailed; as for some lightly heavy-tailed distribution , 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(   > ) in some cases where  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.
Firstly we introduce some definitions on some classes of convolution-equivalent distributions.A distribution  on [0, ∞) belongs to the class of convolution-equivalent distributions, denoted by S(),  ≥ 0, if for any  ∈ R, where  * 2 denotes the convolution of  with itself.More generally, a distribution  on R belongs to the class S(),  ≥ 0, if and only if its right-hand distribution  + () = ()1 {≥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  on R belongs to the class L(),  ≥ 0 if only relation (4) holds.In the case  = 0, we say that L := L(0) is the class of long-tailed distributions.Similarly, a positive function (⋅) is said to be long tailed if lim ( − )/() = 1 for any  ∈ R. Clearly, if a distribution  ∈ L, then its tail probability () is long tailed.Closely related is the class A, which was introduced by Konstantinides et al. [17].A distribution  on R belongs to the class A if  is subexponential, and, for some  > 1, Clearly, all distributions in the classes A, S, and L are heavy tailed.A distribution  of r.v. is said to be heavy tailed if E  = ∞ for any  > 0; otherwise it is said to be light tailed.
For each  ≥ 1, denote the distribution of   ∏  =1   by   , by convention,  =  1 .Now we state our main result as follows.

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 distributions  and  with () > 0 and () > 0 for all  ≥ 0, relation (7) holds for each  > 0, if and only if there is a nonnegative function (⋅) such that Lemma 5. Consider the product .The distribution  of the product belongs to the class A if and only if  ∈ A and relation (7) holds for all  > 0.
Tang [19] obtained an interesting result to show that a light-tailed random variable can be transferred into a heavytailed one through multiplier.Lemma 6.Consider the product  with  ∈ S() for some  > 0 and x = ∞.If relation (7) holds for all  > 0, then  ∈ S.
The last lemma can be found in, for example, Theorem 3.14 of Foss et al. [21].
Lemma 7. Let a reference distribution  on R belong to the class S. Assume that distributions  1 , . . .,   on R satisfy that, for each  = 1, . . ., , the function  +   is long tailed and   () = (()).Then, it holds that Proof of Theorem 1.Now we begin to prove the main result of Theorem 1.
For each  ≥ 1, write where  = stands for equality in distribution.Since  ∈ S() ⊂ L(),  > 0, its tail distribution  is rapidly varying in the sense that lim  () By Lemma 6, we get that  ∈ S. Further, by Lemma 4, there exists a nonnegative function (⋅) such that (9) holds.Thus, by ( 9) and ( 12), for any  > 1, which, together with  ∈ S, implies that  ∈ A.
We proceed to prove relation (8) by induction on .Trivially, the distribution  1 =  of  1 or  1 belongs to the class A, and relation (8) holds for  = 1.Assume that   ∈ A and (8) holds for .We aim to prove that  +1 ∈ A and (8) holds for  + 1, which, by (11), is equivalent to First of all, according to Lemma 2.17 of Foss et al. [21] and   ∈ A ⊂ L, we have, that for any  > 0, which, together with  ∈ S(),  > 0, implies that By (16) For the above-mentioned nonnegative function (⋅), from ( 9) and ( 18 where the last step used the fact that () = ( +1 ()), because, for any  ≥ 0, from which and   ∈ A, Lemma 5 gives that  +1 ∈ A. This completes the proof of Theorem 1.