Electronic Communications in Probability

On free stable distributions

Takahiro Hasebe and Alexey Kuznetsov

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We investigate analytical properties of free stable distributions and discover many connections with their classical counterparts. Our main result is an explicit formula for the Mellin transform, which leads to explicit series representations for the characteristic function and for the density of a free stable distribution. All of these formulas bear close resemblance to the corresponding expressions for classical stable distributions. As further applications of our results, we give an alternative proof of the duality law due to Biane and a new factorization of a classical stable random variable into an independent (in the classical sense) product of a free stable random variable and a power of a Gamma(2) random variable.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 56, 12 pp.

Accepted: 19 August 2014
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L54: Free probability and free operator algebras
Secondary: 60E07: Infinitely divisible distributions; stable distributions

free stable distribution stable distribution Mellin transform

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Hasebe, Takahiro; Kuznetsov, Alexey. On free stable distributions. Electron. Commun. Probab. 19 (2014), paper no. 56, 12 pp. doi:10.1214/ECP.v19-3443. https://projecteuclid.org/euclid.ecp/1465316758

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