A necessary and sufficient condition for the subexponentiality of product distribution

TL;DR

This paper establishes a precise necessary and sufficient condition for the subexponentiality of the product distribution of two independent nonnegative random variables, extending previous sufficient conditions and applying results to insurance risk models.

Contribution

It provides a complete characterization of subexponentiality for product distributions, filling a gap in the understanding of tail behavior in such convolutions.

Findings

Derived a necessary and sufficient condition for subexponentiality of product distribution.

Established conditions for subexponentiality of the original distribution given the product.

Applied results to asymptotic ruin probability in insurance risk models.

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.