Berry-Esseen and Edgeworth approximations for the tail of an infinite sum of weighted gamma random variables

Abstract

Consider the sum $Z={\sum }_{n=1}^{\infty }{\lambda }_n({\eta }_n-\mathbb{E}{\eta }_n)$, where ${\eta }_n$ are i.i.d.~gamma random variables with shape parameter $r>0$, and the ${\lambda }_n$'s are predetermined weights. We study the asymptotic behavior of the tail ${\sum }_{n=M}^{\infty }{\lambda }_n({\eta }_n-\mathbb{E}{\eta }_n)$ which is asymptotically normal under certain conditions. We derive a Berry-Essen bound and Edgeworth expansions for its distribution function. We illustrate the effectiveness of these expansions on an infinite sum of weighted chi-squared distributions.

19 pages, 2 figures