HCM property and the half-Cauchy distribution
Słowa kluczowe:Half-Cauchy distribution, complete monotonicity, generalized gamma convolution, hyperbolically completely monotone function, Pick function, positive stable density
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 /~Zαβ 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.