HCM Property and the Half-Cauchy Distribution

2014/02/05 by Bosch, Pierre · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1402.1059

Abstract

Let Z_\al be a positive α-stable random variable and T_\al=(Z_\al/ Z_\al)^\al, with independents components in the quotient. It is known that T_\al is distributed as the positive branch of a Cauchy random variable with drift. We show that the density of the power transformation T_\al^β is hyperbolically completely monotone in the sense of Thorin and Bondesson if and only if \al≤1/2 and |β|≥ 1/(1-\al). This clarifies a conjecture of Bondesson (1992) on positive stable densities.

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