On free stable distributions
Alexey Kuznetsov (York University)
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.
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Pages: 1-12
Publication Date: August 19, 2014
DOI: 10.1214/ECP.v19-3443
References
- O. Arizmendi; V. Perez-Abreu. On the non-classical infinite divisibility of power semicircle distributions. Commun. Stoch. Anal. 4 (2010), no. 2, 161--178. MR2662723
- O.E. Barndorff-Nielsen; S. Thorbjornsen. A connection between free and classical infinite divisibility. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2004), no. 4, 573--590. MR2105912
- H. Bercovici; V. Pata. Stable laws and domains of attraction in free probability theory. With an appendix by P. Biane. Ann. of Math. (2) 149 (1999), no. 3, 1023--1060. MR1709310
- H. Bercovici; D. Voiculescu. Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42 (1993), no. 3, 733--773. MR1254116
- N. Cressie. A note on the behaviour of the stable distributions for small index $\alpha $. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975/76), no. 1, 61--64. MR0380928
- N. Demni. Kanter random variable and positive free stable distributions. Electron. Commun. Probab. 16 (2011), 137--149. MR2783335
- U. Haagerup; S. Möller. The law of large numbers for the free multiplicative convolution. In Operator Algebra and Dynamics, volume 58 of Springer Proceedings in Mathematics & Statistics, pages 157 -- 186. Springer, Berlin, Heidelberg, 2013.
- A. M. Mathai; R. K. Saxena; H. J. Haubold. The H-Function: Theory and Applications. Springer, 2010.
- V. Pata. Levy type characterization of stable laws for free random variables. Trans. Amer. Math. Soc. 347 (1995), no. 7, 2457--2472. MR1311913
- C. Profeta; T. Simon. Persistence of integrated stable processes. arXiv:1403.1064, 2014.
- V.M. Zolotarev. One-dimensional stable distributions. Translated from the Russian by H. H. McFaden. Translation edited by Ben Silver. Translations of Mathematical Monographs, 65. American Mathematical Society, Providence, RI, 1986. x+284 pp. ISBN: 0-8218-4519-5 MR0854867
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