open access

https://hdl.handle.net/2115/76757

Some properties of the free stable distributions

英語

Wright function

2020

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

56(1)

296-325

We investigate certain analytical properties of the free α-stable densities on the line. We prove that they are all classically infinitely divisible when α≤1 and that they belong to the extended Thorin class when α≤3/4. The Lévy measure is explicitly computed for α=1, showing that free 1-stable distributions are not in the Thorin class except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when α>1. In the one-sided case we prove, refining unimodality, that the densities are whale-shaped, that is their successive derivatives vanish exactly once on their support. We also derive several fine properties of spectrally one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, and several intrinsic features of whale-shaped functions.

学術雑誌論文

Version of Record

0246-0203