A permutation model for free random variables and its classical analogue

Florent Benaych-Georges and Ion Nechita

Vol. 242 (2009), No. 1, 33–51

DOI: 10.2140/pjm.2009.242.33

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