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

Let XX and YY 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 XX and YY on the tail behavior of XYXY. We quantify the impact as the limit, as x→∞x→∞, of the quotient of Pr(XY>x)Pr(XY>x) and Pr(X∗Y∗>x)Pr(X∗Y∗>x), where X∗X∗ and Y∗Y∗ are independent random variables identically distributed as XX and YY, respectively. We obtain an explicit expression for this limit when XX is regularly varying or rapidly varying tailed.