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., Related to D(x) is the difference R(x) between the tail of the convolution and the sum of the tails. 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.