Geodesic
The Difference Between the Product and the Convolution Product of Distribution Functions in $\mathbb{R}^n$
Publications de l'Institut Mathématique, _N_S_89 (2011) no. 103, p. 19 .

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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.
Classification : 26A12 26B99 60E99 60K99
Keywords: subexponential distribution, regular variation, $O$-regularly varying functions, sums of random vectors 
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     author = {E. Omey and R. Vesilo},
     title = {The {Difference} {Between} the {Product} and the {Convolution} {Product} of {Distribution} {Functions} in $\mathbb{R}^n$},
     journal = {Publications de l'Institut Math\'ematique},
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     publisher = {mathdoc},
     volume = {_N_S_89},
     number = {103},
     year = {2011},
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E. Omey; R. Vesilo. The Difference Between the Product and the Convolution Product of Distribution Functions in $\mathbb{R}^n$. Publications de l'Institut Mathématique, _N_S_89 (2011) no. 103, p. 19 . http://geodesic.mathdoc.fr/item/PIM_2011_N_S_89_103_a2/