Classical and free infinite divisibility for Boolean stable laws
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- by Octavio Arizmendi and Takahiro Hasebe PDF
- Proc. Amer. Math. Soc. 142 (2014), 1621-1632 Request permission
Abstract:
We completely determine the free infinite divisibility for the Boolean stable law which is parametrized by a stability index $\alpha$ and an asymmetry coefficient $\rho$. We prove that the Boolean stable law is freely infinitely divisible if and only if one of the following conditions holds: $0<\alpha \leq \frac {1}{2}$; $\frac {1}{2}<\alpha \leq \frac {2}{3}$ and $2-\frac {1}{\alpha }\leq \rho \leq \frac {1}{\alpha }-1$; $\alpha =1,~\rho =\frac {1}{2}$. Positive Boolean stable laws corresponding to $\rho =1$ and $\alpha \leq \frac {1}{2}$ have completely monotonic densities and they are both freely and classically infinitely divisible. We also show that continuous Boolean convolutions of positive Boolean stable laws with different stability indices are also freely and classically infinitely divisible. Boolean stable laws, free stable laws and continuous Boolean convolutions of positive Boolean stable laws are non-trivial examples whose free divisibility indicators are infinity. We also find that the free multiplicative convolution of Boolean stable laws is again a Boolean stable law.References
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Additional Information
- Octavio Arizmendi
- Affiliation: Universität des Saarlandes, FR 6.1–Mathematik, 66123 Saarbrücken, Germany
- Address at time of publication: Research Center for Mathematics, CIMAT, Apartado Postal 402, Guanajuato, GTO 36000, Mexico
- Email: octavius@cimat.mx
- Takahiro Hasebe
- Affiliation: Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
- Address at time of publication: Université de Franche-Comté, 16 route de Gray, 25030 Besançon cedex, France
- MR Author ID: 843606
- Email: thasebe@univ-fcomte.fr
- Received by editor(s): June 2, 2012
- Published electronically: February 13, 2014
- Additional Notes: The first author was supported by DFG-Deutsche Forschungsgemeinschaft Project SP419/8-1
The second author was supported by the Global COE program “Fostering top leaders in mathematics—broadening the core and exploring new ground” at Kyoto University. - Communicated by: Mark M. Meerschaert
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1621-1632
- MSC (2010): Primary 46L54, 60E07
- DOI: https://doi.org/10.1090/S0002-9939-2014-12111-3
- MathSciNet review: 3168468