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Classical scale mixtures of Boolean stable laws


Authors: Octavio Arizmendi and Takahiro Hasebe
Journal: Trans. Amer. Math. Soc. 368 (2016), 4873-4905
MSC (2010): Primary 46L54, 60E07
DOI: https://doi.org/10.1090/tran/6792
Published electronically: October 8, 2015
MathSciNet review: 3456164
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Abstract: We study Boolean stable laws, $ \mathbf {b}_{\alpha ,\rho }$, with stability index $ \alpha $ and asymmetry parameter $ \rho $. We show that the classical scale mixture of $ \mathbf {b}_{\alpha ,\rho }$ coincides with a free mixture and also a monotone mixture of $ \mathbf {b}_{\alpha ,\rho }$. For this purpose we define the multiplicative monotone convolution of probability measures, one supported on the positive real line and the other arbitrary.

We prove that any scale mixture of $ \mathbf {b}_{\alpha ,\rho }$ is both classically and freely infinitely divisible for $ \alpha \leq 1/2$ and also for some $ \alpha >1/2$. Furthermore, we show the multiplicative infinite divisibility of $ \mathbf {b}_{\alpha ,1}$ with respect to classical, free and monotone convolutions.

Scale mixtures of Boolean stable laws include some generalized beta distributions of the second kind, which turn out to be both classically and freely infinitely divisible. One of them appears as a limit distribution in multiplicative free laws of large numbers studied by Tucci, Haagerup and Möller.

We use a representation of $ \mathbf {b}_{\alpha ,1}$ as the free multiplicative convolution of a free Bessel law and a free stable law to prove a conjecture of Hinz and Młotkowski regarding the existence of the free Bessel laws as probability measures. The proof depends on the fact that $ \mathbf {b}_{\alpha ,1}$ has free divisibility indicator 0 for $ 1/2<\alpha $.


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Additional Information

Octavio Arizmendi
Affiliation: Department of Probability and Statistics, CIMAT, Guanajuato, Mexico
Email: octavius@cimat.mx

Takahiro Hasebe
Affiliation: Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kitaku, Sapporo 060-0810, Japan
Email: thasebe@math.sci.hokudai.ac.jp

DOI: https://doi.org/10.1090/tran/6792
Keywords: Free convolution, Boolean stable law, infinite divisibility, mixtures, free Bessel law
Received by editor(s): May 27, 2014
Published electronically: October 8, 2015
Additional Notes: The second author was supported by Marie Curie Actions – International Incoming Fellowships (Project 328112 ICNCP) at University of Franche-Comté and also by the Global COE program “Fostering top leaders in mathematics – broadening the core and exploring new ground” at Kyoto University.
Article copyright: © Copyright 2015 American Mathematical Society