Electronic Communications in Probability

Kanter random variable and positive free stable distributions

Nizar Demni

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Stable laws free probability Fox H-function

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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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