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.