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A Note on the Tail Behavior of Randomly Weighted Sums with Convolution-Equivalently Distributed Random Variables
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.
1. Introduction and Main Result
Let be a sequence of independent and identically distributed (i.i.d.) real-valued random variables with common distribution , and let be another sequence of i.i.d. nonnegative r.v.s with common distribution and right endpoint . Assume that are independent of . In this paper, we are interested in the randomly weighted sum This is because the study for the tail probability can be directly applied to risk theory. Consider a discrete-time insurance risk model. Within period , , 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 . In the terminology of Norberg , the sequences and 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 , . As usual, the probability of ruin by time can be defined by where is interpreted as the initial capital reserve of an insurance company. Clearly, for each , where denotes the positive part of , . If we can establish an asymptotic formula for while doing so does not require , then the same asymptotic formula should hold for the right-hand side of (3) as well. In this way the ruin probability has the same asymptotic behavior as that of the tail probability 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 heavy-tailed 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 , Zhang et al. , Shen et al. , Chen and Yuen , Gao and Wang , and Yi et al.  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 , Hashorva et al. , Yang et al. , Yang and Hashorva , and Yang and Wang  among others. We pointed out that Tang and Tsitsiashvili  achieved some interesting results on the asymptotics for the tail probability 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.
Hereafter, all the limit relationships hold for tending to infinity. For two positive functions and , we write if ; write if ; and write if .
Firstly we introduce some definitions on some classes of convolution-equivalent distributions. A distribution on belongs to the class of convolution-equivalent distributions, denoted by , , if for any , where denotes the convolution of with itself. More generally, a distribution on belongs to the class , , if and only if its right-hand distribution belongs to this class; see Corollary 2.1 of Pakes . The class is called the class of subexponential distributions. A distribution on belongs to the class , if only relation (4) holds. In the case , we say that is the class of long-tailed distributions. Similarly, a positive function is said to be long tailed if for any . Clearly, if a distribution , then its tail probability is long tailed. Closely related is the class , which was introduced by Konstantinides et al. . A distribution on belongs to the class if is subexponential, and, for some , Clearly, all distributions in the classes , , and are heavy tailed. A distribution of r.v. is said to be heavy tailed if for any ; otherwise it is said to be light tailed.
For each , denote the distribution of by , by convention, . Now we state our main result as follows.
Theorem 1. If for some , , and, for all , then, for each ,
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 .
Lemma 4. For two distributions and with and for all , relation (7) holds for each , 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 if and only if and relation (7) holds for all .
Tang  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 product with for some and . If relation (7) holds for all , then .
The last lemma can be found in, for example, Theorem 3.14 of Foss et al. .
Lemma 7. Let a reference distribution on belong to the class . Assume that distributions on satisfy that, for each , 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 , write where stands for equality in distribution. Since , , its tail distribution is rapidly varying in the sense that By Lemma 6, we get that . Further, by Lemma 4, there exists a nonnegative function such that (9) holds. Thus, by (9) and (12), for any , which, together with , implies that .
We proceed to prove relation (8) by induction on . Trivially, the distribution of or belongs to the class , and relation (8) holds for . Assume that and (8) holds for . We aim to prove that and (8) holds for , which, by (11), is equivalent to
First of all, according to Lemma 2.17 of Foss et al.  and , we have, that for any , which, together with , implies that By (16) and , we have that for any , which shows that the function is long tailed. Since and are independent of each other, thus, by , we can apply Lemma 7 to derive from the induction assumption and (16) that For the above-mentioned nonnegative function , from (9) and (18), we obtain that where the last step used the fact that , because, for any , and by . Relation (19) means that (14) holds.
Finally, by (9) and (20), we have that from which and , Lemma 5 gives that .
This completes the proof of Theorem 1.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are most grateful to the two referees and the editor for their very thorough reading of the paper and valuable suggestions to improve the presentation of this paper. This research is supported by the National Natural Science Foundation of China (nos. 11001052 and 71171046), China Postdoctoral Science Foundation (no. 2012M520964), the Natural Science Foundation of Jiangsu Province of China (no. BK20131339), Qing Lan Project, and Project of Construction for Superior Subjects of Statistics & Audit Science and Technology of Jiangsu Higher Education Institutions.
Q. Tang and G. Tsitsiashvili, “Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks,” Stochastic Processes and Their Applications, vol. 108, no. 2, pp. 299–325, 2003.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. Yi, Y. Chen, and C. Su, “Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 365–372, 2011.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet