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Some properties of the free stable distributions

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Title: Some properties of the free stable distributions
Authors: Hasebe, Takahiro Browse this author →KAKEN DB
Simon, Thomas Browse this author
Wang, Min Browse this author
Keywords: Free stable distribution
Infinite divisibility
Shape of densities
Wright function
Issue Date: 2020
Journal Title: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Volume: 56
Issue: 1
Start Page: 296
End Page: 325
Publisher DOI: 10.1214/19-AIHP962
Abstract: 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.
Type: article
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)

Submitter: 長谷部 高広

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