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

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 XY . We quantify the impact as the limit, as x → ∞, of the quotient of Pr(XY > x) and Pr(XY ∗ > 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. © 2011 Elsevier B.V. All rights reserved.