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

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. nema