The Power Problem for Generalized Gamma Convolutions (GGC) and Related Questions

TL;DR

This paper explores the closure properties of generalized gamma convolutions (GGC), proving they are also closed under q-th powers for q>1, and applies this to sums and products of gamma variables.

Contribution

The paper establishes that GGCs are closed under q-th powers (q>1), a new property not previously known, expanding understanding of GGC operations.

Findings

GGC is closed under q-th powers for q>1.

Explicit formulas for sums of gamma variables are used.

Applications to sums and products of gamma variables and symmetrized GGCs.

Abstract

The class of generalized gamma convolutions (GGC) is closed with respect to (wrt) change of scales, weak limits and addition and multiplication of independent random variables. Our main result adds the new property that GGC is also closed wrt q-th powers, q>1. The proof uses explicit formulas for the densities of finite sums of independent gamma variables, hyperbolically completely monotone functions (HCM) and the Laplace transform. The result is applied to sums and products of independent gamma variables and to symmetric extended GGC (symEGGC).