The Arithmetic of Distributions in Free Probability Theory

TL;DR

This paper develops an analytical framework for free convolutions in probability distributions, characterizes their factorization properties, and shows the density of indecomposable elements within the semigroup.

Contribution

It introduces an analytical approach to free convolutions using Nevanlinna and Schur functions, and proves a Khintchine type theorem for distribution factorization.

Findings

Characterization of free convolution semigroup structure.

Existence of indecomposable ('prime') factors in distributions.

Density of indecomposable elements in the semigroup.

Abstract

We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and of Schur functions. We consider the set of probability distributions as a semigroup $\bold M$ equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of $\bold M$ contains either indecomposable ("prime") factors or it belongs to a class, say $I_0$, of distributions without indecomposable factors. In contrast to the classical convolution semigroup in the free additive and multiplicative convolution semigroups the class $I_0$ consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in $\bold M$.