On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC)

Let $Y$ be a standard Gamma(k) distributed random variable, $k>0$, and let $X$ be an independent positive random variable. We prove that if $X$ has a hyperbolically monotone density of order $k$ ($H{M}_k$), then the distributions of $Y\cdot X$ and $Y\mathrm{/}X$ are generalized gamma convolutions (GGC). This result extends results of Roynette et al. and Behme and Bondesson, who treated respectively the cases $k=1$ and $k$ an integer. We give a proof that covers all $k>0$ and gives explicit formulas for the relevant functions that extend those found by Behme and Bondesson in the integer case.