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arXiv · 1605.02319

A necessary and sufficient condition for the subexponentiality of product distribution

Hui Xu·Fengyang Cheng·Yuebao Wang·Dongya Cheng
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Abstract

Let X and Y be two independent and nonnegative random variables with corresponding distributions F and G. Denote by H the distribution of the product XY , called the product convolution of F and G. Cline and Samorodnitsky (1994) proposed sufficient conditions for H to be subexponential, given the subexponentiality of F. Relying on a related result of Tang (2008) on the long-tail of product convolution, we obtain a necessary and sufficient condition for the subexponentiality of H, given that of F. We also study the reverse problem and obtain sufficient conditions for the subexponentiality of F given that of H. Finally, we apply the obtained results to the asymptotic study of the ruin probability in a discrete-time insurance risk model with stochastic returns.

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Hui Xu, Fengyang Cheng, Yuebao Wang, Dongya Cheng. 2017-10-01. A necessary and sufficient condition for the subexponentiality of product distribution. https://arxiv.org/abs/1605.02319

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