Vol. 242, No. 1, 2009

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ISSN: 0030-8730
A permutation model for free random variables and its classical analogue

Florent Benaych-Georges and Ion Nechita

Vol. 242 (2009), No. 1, 33–51
Abstract

In this paper, we generalize a permutation model for free random variables which was first proposed by Biane. We also construct its classical probability analogue, by replacing the group of permutations with the group of subsets of a finite set endowed with the symmetric difference operation. These constructions provide new discrete approximations of the respective free and classical Wiener chaos. As a consequence, we obtain explicit examples of nonrandom matrices which are asymptotically free or independent. The moments and the free (respectively classical) cumulants of the limiting distributions are expressed in terms of a special subset of (noncrossing) pairings. At the end of the paper we present some combinatorial applications of our results.

Keywords
free probability, limit theorem, combinatorics of partitions, stochastic measure
Mathematical Subject Classification 2000
Primary: 46L54, 60F05
Secondary: 05A18, 05A05
Milestones
Received: 28 January 2008
Accepted: 26 January 2009
Published: 1 September 2009
Authors
Florent Benaych-Georges
LPMA
UPMC Univ Paris 6
Case courier 188
4, Place Jussieu
75252 Paris Cedex 05
France
http://www.cmapx.polytechnique.fr/~benaych/
Ion Nechita
Université Claude Bernard Lyon 1
Institut Camille Jordan
CNRS UMR 5208
43, Boulevard du 11 novembre 1918
69622 Villeurbanne Cedex
France
http://math.univ-lyon1.fr/~nechita/