THE DIFFERENCE BETWEEN THE PRODUCT AND THE CONVOLUTION PRODUCT OF DISTRIBUTION FUNCTIONS IN $mathbb{R}^n$

摘要

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