Extensions of Breiman’s Theorem of Product of Dependent Random Variables with Applications to Ruin Theory

Abstract

We consider the tail behavior of the product of two dependent random variables X and $\Theta $. Motivated by Denisov and Zwart (J Appl Probab 44:1031–1046, 2007), we relax the condition of the existing $\alpha \,+\,\epsilon $ th moment of $\Theta $ in Breiman’s theorem to the existing $\alpha $th moment and obtain the similar result as Breiman’s theorem of the dependent product $X \Theta $, while X and $\Theta $ follow a copula function. As applications, we consider a discrete-time insurance risk model with dependent insurance and financial risks and derive the asymptotic tail behaviors for the (in)finite-time ruin probabilities.

Keywords

Copula / Dependent product / Regular variation / Ruin probabilities

Funding

National Natural Science Foundation of China(71771203)

National Key Research and Development Plan(2016YFC0800104)