The Difference between the Product and the Convolution Product of Distribution Functions in R

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.