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WROCŁAW UNIVERSITY
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TECHNOLOGY

Contents of PMS, Vol. 41, Fasc. 1,
pages 1 - 7
DOI: 10.37190/0208-4147.41.1.1
Published online 1.4.2021
 

On mixtures of gamma distributions, distributions with hyperbolically monotone densities and generalized gamma convolutions (GGC)} hyperbolically monotone densities and generalized gamma convolutions (GGC)

Tord Sjödin

Abstract: Let \(Y\) be a standard \({\rm Gamma}(k)\) distributed random variable (rv), \(k>0\), and let \(X\) be an independent positive rv. If \(X\) has a hyperbolically monotone density of order \(k\) (\({\rm HM}_k\)), then \(Y\cdot X\) and \(Y/X\) are generalized gamma convolutions (GGC). This extends work by Roynette et al. and Behme and Bondesson. The same conclusion holds with \(Y\) replaced by a finite sum of independent gamma variables with sum of shape parameters at most \(k\). Both results are applied to subclasses of GGC.

2010 AMS Mathematics Subject Classification: Primary 60E10; Secondary 62E15.

Keywords and phrases: gamma distribution, hyperbolically monotone function, Laplace transform, generalized gamma convolution (GGC).

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