Revisiting the Product of Random Variables

Abstract

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.

On the non-closure under convolution for strong subexponential distributions

“…We consider the non-negative claim sizes, where it was possible to avoid condition (4.1). It worth to mention that recently several attempts to study this problem under dependent random variables, as for example in [5], [35], [59] among others. Furthermore, in [32] we find for random variable X, whose support is the whole real axis R, that class OA manifests closure property with respect to convolution product under the condition (4.1), from this aspect, the second part of Theorem 4.1 extends this result in non-negative case.…”