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

Tord Sjodin

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

Abstract

Tord Sjodin

The class of generalized gamma convolutions (GGC) is closed with respect to change of scale, weak limits and addition and multiplication of independent random variables. Our main result confirms an old conjecture that GGC is also closed wrt q− th powers, q > 1. The proof uses explicit iterative formulas for the densities of finite sums of independent gamma variables, hyperbolically completely monotone functions (HCM) and the Laplace transform. We apply the result to sums and products of q− th powers of independent GGCs, q ≥ 1, symmetric extended GGC (symEGGC) and a new proof that X ∼ GGC implies Exp(X) ∼ GGC.

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