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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.