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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Classical and free infinite divisibility for Boolean stable laws


Authors: Octavio Arizmendi and Takahiro Hasebe
Journal: Proc. Amer. Math. Soc. 142 (2014), 1621-1632
MSC (2010): Primary 46L54, 60E07
Published electronically: February 13, 2014
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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.


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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
Email: thasebe@univ-fcomte.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-12111-3
PII: S 0002-9939(2014)12111-3
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.