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On a class of explicit Cauchy–Stieltjes transforms related to monotone stable and free Poisson laws
| Title: | On a class of explicit Cauchy–Stieltjes transforms related to monotone stable and free Poisson laws |
| Authors: | Arizmendi, Octavio Browse this author | | Hasebe, Takahiro Browse this author →KAKEN DB |
| Keywords: | beta distribution | | free infinite divisibility | | free Poisson law | | monotone stable law |
| Issue Date: | 2013 |
| Journal Title: | Bernoulli |
| Volume: | 19 |
| Issue: | 5B |
| Start Page: | 2750 |
| End Page: | 2767 |
| Publisher DOI: | 10.3150/12-BEJ473 |
| Abstract: | We consider a class of probability measures $μ^{α}_{s,r}$ which have explicit Cauchy–Stieltjes transforms. This class includes a symmetric beta distribution, a free Poisson law and some beta distributions as special cases. Also, we identify $μ^{α}_{s,2}$ as a free compound Poisson law with Lévy measure a monotone α-stable law. This implies the free infinite divisibility of $μ^{α}_{s,2}$. Moreover, when symmetric or positive, $μ^{α}_{s,2}$ has a representation as the free multiplication of a free Poisson law and a monotone α-stable law. We also investigate the free infinite divisibility of $μ^{α}_{s,r}$ for r≠2. Special cases include the beta distributions $B(1−\frac{1}{r},1+\frac{1}{r})$ which are freely infinitely divisible if and only if 1≤r≤2. |
| Type: | article |
| URI: | http://hdl.handle.net/2115/75993 |
| 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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