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

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 L\'evy processes are mentioned.