HCM property and the half-Cauchy distribution

  1. Pierre Bosch


Let Zα and ~Zβ be two independent positive -stable random variables. It is known that (Z α/~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 /~Z α) 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.

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Probability and Mathematical Statistics

35, z. 2, 2015

Pages from 191 to 200

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