A novel single-gamma approximation to the sum of independent gamma variables, and a generalization to infinitely divisible distributions

Abstract

It is well known that the sum $S$ of $n$ independent gamma variables—which occurs often, in particular in practical applications—can typically be well approximated by a single gamma variable with the same mean and variance (the distribution of $S$ being quite complicated in general). In this paper, we propose an alternative (and apparently at least as good) single-gamma approximation to $S$. The methodology used to derive it is based on the observation that the jump density of $S$ bears an evident similarity to that of a generic gamma variable, $S$ being viewed as a sum of $n$ independent gamma processes evaluated at time $1$. This observation motivates the idea of a gamma approximation to $S$ in the first place, and, in principle, a variety of such approximations can be made based on it. The same methodology can be applied to obtain gamma approximations to a wide variety of important infinitely divisible distributions on $\mathbb{R}_{+}$ or at least predict/confirm the appropriateness of the moment-matching method (where the first two moments are matched); this is demonstrated neatly in the cases of negative binomial and generalized Dickman distributions, thus highlighting the paper’s contribution to the overall topic.