Revisiting the product of random variables

For a large class of distribution functions we study properties of the product of random variables X and Y. We take into account the dependency structure between X and Y by making assumptions about the asymptotic equality P(X > x|Y = y) ~ h(y)P(X > x) as x → ∞, uniformly for y in the range of Y. As particular consequences, some well-known results concerning the product of random variables are reviewed, among them the Breiman’s theorem. An application is made in the case where the dependence between X and Y is characterized by asymptotic conditions on their copula.