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A necessary and sufficient condition for the subexponentiality of the product convolution. (English) Zbl 1443.60018

Summary: 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 \(G\). D. B. H. Cline and G. Samorodnitsky [Stochastic Processes Appl. 49, No. 1, 75–98 (1994; Zbl 0799.60015)] proposed sufficient conditions for \(H\) to be subexponential, given the subexponentiality of \(F\). Relying on a related result of Q. Tang [Extremes 11, No. 4, 379–391 (2008; Zbl 1199.60040)] on the long-tail of the 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.

MSC:

60E05 Probability distributions: general theory
91G40 Credit risk
62E20 Asymptotic distribution theory in statistics
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