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Some properties of the free stable distributions
| 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 |
| URI: | http://hdl.handle.net/2115/76757 |
| 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)
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Submitter: 長谷部 高広
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