Multilinear function series and transforms in free probability theory

TL;DR

This paper introduces algebraic structures and transforms in free probability theory, extending classical tools to handle all moments of noncommutative random variables through multilinear function series.

Contribution

It develops the algebra of multilinear function series and defines unsymmetrized R- and T-transforms, broadening the scope of free probability analysis.

Findings

Introduces algebra Mul[[B]] and SymMul[[B]] for multilinear functions.

Defines unsymmetrized R- and T-transforms for noncommutative variables.

Uses noncrossing linked partitions to analyze properties of the T-transform.

Abstract

The algebra Mul[[B]] of formal multilinear function series over an algebra B and its quotient SymMul[[B]] are introduced, as well as corresponding operations of formal composition. In the setting of Mul[[B]], the unsymmetrized R- and T-transforms of random variables in B-valued noncommutative probability spaces are introduced. These satisfy properties analogous to the usual R- and T-transforms, (the latter being just the reciprocal of the S-transform), but describe all moments of a random variable, not only the symmetric moments. The partially ordered set of noncrossing linked partitions is introduced and is used to prove properties of the unsymmetrized T-transform.