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

TL;DR

This paper investigates the tail behavior of an infinite sum of weighted gamma variables, deriving Berry-Essen bounds and Edgeworth expansions to approximate its distribution, with applications to weighted chi-squared sums.

Contribution

It provides new Berry-Essen bounds and Edgeworth expansions for the tail of an infinite sum of weighted gamma variables, enhancing approximation accuracy.

Findings

The tail sum is asymptotically normal under certain conditions.

Derived explicit Berry-Essen bounds for the tail distribution.

Developed Edgeworth expansions improving distribution approximation.

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.