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

Consider the sum $Z = \sum_{n=1}^\infty λ_n (η_n - \mathbb{E}η_n)$, where $η_n$ are i.i.d.~gamma random variables with shape parameter $r > 0$, and the $λ_n$'s are predetermined weights. We study the asymptotic behavior of the tail $\sum_{n=M}^\infty λ_n (η_n - \mathbb{E}η_n)$ 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.