A permutation model for free random variables and its classical analogue

Details A permutation model for free random variables and its classical analogue A permutation model for free random variables and its classical analogue

2008-01-28

arxiv.org/abs/0801.4229v3

Mathematics

Probability

13 pages, to appear in Pacific Journal of Mathematics

Scientific paper

In this paper, we generalize a permutation model for free random variables which was first proposed by Biane in \cite{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 non random matrices which are asymptotically free or independent. The moments and the free (resp. 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.

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