A class of scale mixtures of gamma(k)-distributions that are generalized gamma convolutions

Abstract

Let $k>0$ be an integer and $Y$ a standard Gamma$(k)$ distributed random variable. Let $X$ be an independent positive random variable with a density that is hyperbolically monotone (HM) of order $k.$ Then $Y\cdot X$ and $Y/X$ both have distributions that are generalized gamma convolutions (GGCs). This result extends a result of Roynette et al. from 2009 who treated the case $k=1$ but without use of the HM-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of Lévy processes are mentioned.

References in corpus (2)

• Further examples of GGC and HCM densities
• A proof of Bondesson's conjecture on stable densities