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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 | | Simon, Thomas Browse this author | | Wang, Min Browse this author |
| Keywords: | Free stable distribution | | Infinite divisibility | | Shape of densities | | Wright function |
| Issue Date: | Feb-2020 |
| Publisher: | Institute of Mathematical Statistics |
| Journal Title: | Annales de l'I.H.P. 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 alpha-stable densities on the line. We prove that they are all classically infinitely divisible when alpha < 1 and that they belong to the extended Thorin class when alpha < 3/4. The Levy measure is explicitly computed for alpha = 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 alpha > 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/2115/76963 |
| 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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