# The arithmetic of distributions in free probability theory

Open Mathematics (2011)

• Volume: 9, Issue: 5, page 997-1050
• ISSN: 2391-5455

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## Abstract

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We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup 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 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 M.

## How to cite

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Gennadii Chistyakov, and Friedrich Götze. "The arithmetic of distributions in free probability theory." Open Mathematics 9.5 (2011): 997-1050. <http://eudml.org/doc/269792>.

abstract = {We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup 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 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 M.},
author = {Gennadii Chistyakov, Friedrich Götze},
journal = {Open Mathematics},
keywords = {Free random variables; Cauchy transforms; Free convolutions; Delphic semigroups; Khintchine’s theorems; free random variables; free convolutions; Khintchine's theorems},
language = {eng},
number = {5},
pages = {997-1050},
title = {The arithmetic of distributions in free probability theory},
url = {http://eudml.org/doc/269792},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Friedrich Götze
TI - The arithmetic of distributions in free probability theory
JO - Open Mathematics
PY - 2011
VL - 9
IS - 5
SP - 997
EP - 1050
AB - We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup 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 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 M.
LA - eng
KW - Free random variables; Cauchy transforms; Free convolutions; Delphic semigroups; Khintchine’s theorems; free random variables; free convolutions; Khintchine's theorems
UR - http://eudml.org/doc/269792
ER -

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