Infinite divisibility of interpolated gamma powers

This paper is concerned with the distribution properties of the binomial a X + b X α , where X is a gamma random variable. We show in particular that a X + b X α is infinitely divisible for all α ∈ [ 1 , 2 ] and a , b ∈ R + , and that for α = 2 the second order polynomial a X + b X 2 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.