Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 19 (2014), paper no. 56, 12 pp.
On free stable distributions
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.
Electron. Commun. Probab., Volume 19 (2014), paper no. 56, 12 pp.
Accepted: 19 August 2014
First available in Project Euclid: 7 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 46L54: Free probability and free operator algebras
Secondary: 60E07: Infinitely divisible distributions; stable distributions
This work is licensed under a Creative Commons Attribution 3.0 License.
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