The Arithmetic of Distributions in Free Probability Theory

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 $\mathbf{M}$ equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of $\mathbf{M}$ 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 $\mathbf{M}$.

66 pages; latex; 5 figures; corrected version of proofs of Khintchine type theorems. For details see end of introduction

References in corpus (1)

• Complex analysis methods in noncommutative probability

Cited by in corpus (2)

• The normal distribution is $⊞$-infinitely divisible
• Monotone and Boolean Convolutions for Non-compactly Supported Probability Measures