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.