Electronic Communications in Probability

Kanter random variable and positive free stable distributions

Nizar Demni

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Abstract

According to a representation due to M. Kanter, the density of some power of a positive stable distribution is a completely monotone function. In this paper, we first derive its representative Bernstein measure which also describes the law of some function of a uniform random variable, referred to below as the Kanter random variable. Then, the distribution function of the latter variable is written down and gives a more explicit description of the non commutative analogue of positive stable distributions in the setting of Voiculescu's free probability theory. Analytic evidences of the occurrence of the Kanter random variable in both the classical and the free settings conclude the exposition.

Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 14, 137-149.

Dates
Accepted: 17 March 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465261970

Digital Object Identifier
doi:10.1214/ECP.v16-1608

Mathematical Reviews number (MathSciNet)
MR2783335

Zentralblatt MATH identifier
1225.60029

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 33E12: Mittag-Leffler functions and generalizations 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Stable laws free probability Fox H-function

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Demni, Nizar. Kanter random variable and positive free stable distributions. Electron. Commun. Probab. 16 (2011), paper no. 14, 137--149. doi:10.1214/ECP.v16-1608. https://projecteuclid.org/euclid.ecp/1465261970


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