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February 2017 A class of scale mixtures of $\operatorname{Gamma}(k)$-distributions that are generalized gamma convolutions
Anita Behme, Lennart Bondesson
Bernoulli 23(1): 773-787 (February 2017). DOI: 10.3150/15-BEJ761


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.


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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.


Received: 1 February 2015; Revised: 1 July 2015; Published: February 2017
First available in Project Euclid: 27 September 2016

zbMATH: 1372.60014
MathSciNet: MR3556792
Digital Object Identifier: 10.3150/15-BEJ761

Keywords: Excursion theory , Exponential functionals , generalized gamma convolution , hyperbolic monotonicity , Lévy process , products and ratios of independent random variables

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 1 • February 2017
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