2018/05/03 by Takahiro Hasebe, Hasebe, Takahiro, Thomas Simon +3
Economics, Econometrics and Finance · Mathematics · #33E20 #46L54 #60E07 #62E15 #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1805.01133
openalex publication_date 2018/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 the free 1-stable random variables are not Thorin 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. Finally, we derive a collection of results connected to the fine structure of the one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, a new identity for the Beta-Gamma algebra, and several intrinsic properties of whale-shaped densities.