Abstract
Let $k>0$ be an integer and $Y$ a standard $\operatorname{Gamma}(k)$ 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 ($\mathrm{GGC}$s). This result extends a result of Roynette et al. from 2009 who treated the case $k=1$ but without use of the $\mathrm{HM}$-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of Lévy processes are mentioned.
Citation
Anita Behme. Lennart Bondesson. "A class of scale mixtures of $\operatorname{Gamma}(k)$-distributions that are generalized gamma convolutions." Bernoulli 23 (1) 773 - 787, February 2017. https://doi.org/10.3150/15-BEJ761
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