A proof of Bondesson's conjecture on stable densities
We show that positive $α$-stable densities are hyperbolically completely monotone if and only if $α\le 1/2$. This gives a positive answer to a question raised by L. Bondesson in 1977.
math.PR↗
arXiv subjects
Pierre Bosch
Publications and source records attributed to Pierre Bosch.
On the infinite divisibility of inverse Beta distributions
We show that all negative powers B_{a,b}^-{s} of the Beta distribution are infinitely divisible. The case b<1 follows by complete monotonicity, the case b > 1, s > 1 by hyperbolically complete monotonicity and the case b > 1, s < 1 by a Lévy perpetuity argument involving the hypergeometric series. We also observe that B_{a,b}^{-s} is self-decomposable whenever 2a + b + s + bs > 1, and that it is not always a generalized Gamma convolution. On the other hand, we prove that all negative powers of the Gamma distribution are generalized Gamma convolutions, answering to a recent question of L. Bondesson.
math.PR↗
HCM Property and the Half-Cauchy Distribution
Let $Z_\al$ be a positive $α$-stable random variable and $T_\al=(Z_\al/\tilde Z_\al)^\al,$ 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 $\al\le1/2$ and $|β|\ge 1/(1-\al).$ This clarifies a conjecture of Bondesson (1992) on positive stable densities.
math.PR↗
On the self-decomposability of the Fréchet distribution
Let $\{Γ_t, \, t\ge 0\}$ be the Gamma subordinator. Using a moment identification due to Bertoin-Yor (2002), we observe that for every $t > 0$ and $α\in (0,1)$ the random variable $Γ_t^{-α}$ is distributed as the exponential functional of some spectrally negative Lévy process. This entails that all size-biased samplings of Fréchet distributions are self-decomposable and that the extreme value distribution $F_ξ$ is infinitely divisible if and only if $ξ\not\in (0,1),$ solving problems raised by Steutel (1973) and Bondesson (1992). We also review different analytical and probabilistic interpretations of the infinite divisibility of $Γ_t^{-α}$ for $t,α> 0.$
math.PR↗