The product of two dependent random variables with regularly varying or rapidly varying tails

Abstract

Let X and Y be two nonnegative and dependent random variables following a generalized Farlie–Gumbel–Morgenstern distribution. In this short note, we study the impact of a dependence structure of X and Y on the tail behavior of X Y . We quantify the impact as the limit, as x → ∞ , of the quotient of Pr ( X Y > x ) and Pr ( X ∗ Y ∗ > x ) , where X ∗ and Y ∗ are independent random variables identically distributed as X and Y , respectively. We obtain an explicit expression for this limit when X is regularly varying or rapidly varying tailed.