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

E. Omey and R. Vesilo

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

Keywords: subexponential distribution, regular variation, $O$-regularly varying functions, sums of random vectors

Classification (MSC2000): 26A12, 26B99, 60E99, 60K99

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