## Bernoulli

• Bernoulli
• Volume 23, Number 1 (2017), 773-787.

### A class of scale mixtures of $\operatorname{Gamma}(k)$-distributions that are generalized gamma convolutions

#### 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.

#### Article information

Source
Bernoulli Volume 23, Number 1 (2017), 773-787.

Dates
Revised: July 2015
First available in Project Euclid: 27 September 2016

https://projecteuclid.org/euclid.bj/1475001374

Digital Object Identifier
doi:10.3150/15-BEJ761

Mathematical Reviews number (MathSciNet)
MR3556792

Zentralblatt MATH identifier
1372.60014

#### Citation

Behme, Anita; Bondesson, Lennart. A class of scale mixtures of $\operatorname{Gamma}(k)$-distributions that are generalized gamma convolutions. Bernoulli 23 (2017), no. 1, 773--787. doi:10.3150/15-BEJ761. https://projecteuclid.org/euclid.bj/1475001374

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