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On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC)

Sjödin Tord

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 1

1--7

Let Y be a standard 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 (HMk), then Y · 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.

HCM property and the half-Cauchy distribution

Bosch P.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 2

191--200

Let Zα and Zα be two independent positive α-stable random variables. It is known that (Zα/Zα)α is distributed as the positive branch of a Cauchy random variable with drift. We show that the density of the power transformation (Zα/Zα)β is hyperbolically completely monotone in the sense of Thorin and Bondesson if and only if α ≤ 1/2 and |β| ≥ α/(1−α). This clarifies a conjecture of Bondesson (1992) on positive stable densities.