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

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).