## Electronic Communications in Probability

### On free stable distributions

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ecp/1465316758

Digital Object Identifier
doi:10.1214/ECP.v19-3443

Mathematical Reviews number (MathSciNet)
MR3254735

Zentralblatt MATH identifier
1317.46049

Rights

#### Citation

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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