The Difference between the product and the convolution product of distribution functions in Rn

Abstract

Assume that X→ and Y→ are independent, nonnegative d-dimensional random vectors with distribution function (d.f.) F(x→) and G(x→), respectively. We are interested in estimates for the difference between the product and the convolution product of F and G, i.e., D(x→) = F(x→)G(x→) − F ∗ G(x→). Related to D(x→) is the difference R(x→) between the tail of the convolution and the sum of the tails: R(x→) = (1 − F ∗ G(x→))−(1 − F(x→) + 1 − G(x→)). We obtain asymptotic inequalities and asymptotic equalities for D(x→) and R(x→). The results are multivariate analogues of univariate results obtained by several authors before.