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Free infinite divisibility for powers of random variables
| Title: | Free infinite divisibility for powers of random variables |
| Authors: | Hasebe, Takahiro Browse this author →KAKEN DB |
| Issue Date: | 2016 |
| Journal Title: | ALEA : Latin American Journal of Probability and Mathematical Statistics |
| Volume: | 13 |
| Issue: | 1 |
| Start Page: | 309 |
| End Page: | 336 |
| Abstract: | We prove that $X^r$ follows a free regular distribution, i.e. the law of a nonnegative free Lévy process if: (1) $X$ follows a free Poisson distribution without an atom at 0 and $r ∈ (−∞, 0] ∪ [1,∞)$; (2) $X$ follows a free Poisson distribution with an atom at 0 and $r ≥ 1$; (3) $X$ follows a mixture of some HCM distributions and $|r| ≥ 1$; (4) $X$ follows some beta distributions and $r$ is taken from some interval. In particular, if $S$ is a standard semicircular element then $|S|^r$ is freely infinitely divisible for $r ∈ (−∞, 0]∪[2,∞)$. Also we consider the symmetrization of the above probability measures, and in particular show that $|S|^r$ sign($S$) is freely infinitely divisible for $r ≥ 2$. Therefore $S^n$ is freely infinitely divisible for every $n ∈ N$. The results on free Poisson and semicircular random variables have a good correspondence with classical ID properties of powers of gamma and normal random variables. |
| Type: | article |
| URI: | http://hdl.handle.net/2115/75968 |
| 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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