  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.

Probability