On the long tail property of product convolution
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Lithuanian Mathematical Journal
On the long tail property of product convolution∗ Zhaolei Cui a and Yuebao Wang b,1 a
School of mathematics and statistics, Changshu Institute of Technology, Suzhou, China, 215000 b School of Mathematical Sciences, Soochow University, Suzhou, China, 215006 (e-mail: [email protected]; [email protected])
Received February 24, 2019; revised November 4, 2019
Abstract. Let X and Y be two independent random variables with corresponding distributions F and G on [0, ∞). The distribution of the product XY , which is called the product convolution of F and G, is denoted by H. In this paper, we give some suitable conditions on F and G, under which the distribution H belongs to the long-tailed distribution class. Here F is a generalized long-tailed distribution, not necessarily an exponential distribution. Finally, we give a series of examples to show that our conditions are satisfied by many distributions, and one of them is necessary in some sense. MSC: primary 60E05; secondary 62E20, 60G50 Keywords: product convolution, long tail property, generalized long-tailed distribution, exponential distribution
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Introduction and main results
In this paper, X and Y are two independent random variables with corresponding distributions F and G on [0, ∞), which means F (x) := 1 − F (x) > 0 for all x 0 and F (x) = 1 for all x < 0, and G is the same. The distribution H of the product XY is called the product convolution of F and G, as opposed to the usual sum convolution F ∗ G defined as the distribution of the sum X + Y . It is well known that the product convolution plays an important role in applied probability. For example, the present value of a random future claim, which is one of most fundamental quantities in finance and insurance, is expressed as the product of the random claim amount and the corresponding stochastic present value factor. Thus the study of the tail behavior of product convolution has immediate implications in finance and insurance. There the product convolution is often required to belong to the subexponential distribution class; see Embrechts and Goldie [11], Cline [7], Cline and Samorodnitsky [8], Tang and Tsitsiashvili [23, 24], Tang [21, 22], Liu and Tang [18], Samorodnitsky and Sun [19], Xu et al. [28]. For this, as Cline and Samorodnitsky [8] points out, “The first one provides conditions for H to be ‘long-tailed’, including the closure results for L (see below for its definition).” This paper focuses on this issue. To this end, we first introduce the concepts and notations of some related classes of distributions V on [0, ∞) or (−∞, ∞). Unless otherwise stated, all limit relations are stated as x → ∞. With s(f, g) := lim sup f (x)/g(x) for two positive functions f and g, we write f (x) = o(g(x)) if s(f, g) = 0, f (x) = O(g(x)) if s(f, g) < ∞, ∗ 1
Research supported by the National Science Foundation of China (No. 11071182). Corresponding author.
c 2020 Springer Science+Business Media, LLC 0363-1672/20/6002-0001
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Z. Cui and Y. Wang
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