Infinite divisibility of interpolated gamma powers

Abstract

This paper is concerned with the distribution properties of the binomial aX+bXα, where X is a gamma random variable. We show in particular that aX+bXα is infinitely divisible for all α∈[1,2] and a,b∈R+, and that for α=2 the second order polynomial aX+bX2 is a generalized gamma convolution whose Thorin density and Wiener–gamma integral representation are computed explicitly. As a byproduct we deduce that fourth order multiple Wiener integrals are in general not infinitely divisible.