The Annals of Probability

A continuous semigroup of notions of independence between the classical and the free one

Florent Benaych-Georges and Thierry Lévy

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In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for noncommutative random variables. These notions are related to the liberation process introduced by Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence.

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Ann. Probab., Volume 39, Number 3 (2011), 904-938.

First available in Project Euclid: 16 March 2011

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Primary: 46L54: Free probability and free operator algebras 15A52

Free probability independence random matrices unitary Brownian motion convolution cumulants


Benaych-Georges, Florent; Lévy, Thierry. A continuous semigroup of notions of independence between the classical and the free one. Ann. Probab. 39 (2011), no. 3, 904--938. doi:10.1214/10-AOP573.

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