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

TL;DR

This paper proves that multiplying or dividing a gamma-distributed variable by a positive independent variable with a hyperbolically monotone density results in a generalized gamma convolution distribution, extending previous specific cases.

Contribution

It generalizes previous results by showing this property holds for all positive shape parameters and provides explicit formulas for the related functions.

Findings

Distributions of Y·X and Y/X are GGC when X has HM_k density.

Extends previous results from specific integer cases to all k>0.

Provides explicit formulas for the functions involved.

Abstract

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 $k$ ($HM_k$), then the distributions of $Y\cdot X$ and $Y/X$ 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.