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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 shapeparameter $r > 0$, and the $\lambda_n$'s are predetermined weights. We studythe asymptotic behavior of the tail $\sum_{n=M}^\infty \lambda_n (\eta_n -\mathbb{E}\eta_n)$ which is asymptotically normal under certain conditions. Wederive a Berry-Essen bound and Edgeworth expansions for its distributionfunction. We illustrate the effectiveness of these expansions on an infinitesum of weighted chi-squared distributions.

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