On Mixtures of Gamma distributions, distributions with hyperbolically monotone densities and Generalized Gamma Convolutions (GGC)

Abstract

Let Y be a standard Gamma(k) distributed random variable (rv), k > 0, and let X be an independent positive rv. If X has a hyperbolically monotone density of order k (HM k ), then Y • X and Y /X are generalized gamma convolutions (GGC). This extends work by Roynette et al. and Behme and Bondesson. The same conclusion holds with Y replaced by a finite sum of independent gamma variables with sum of shape parameters at most k. Both results are applied to subclasses of GGC.