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The arithmetic of distributions in free probability theory

100%

Chistyakov G. , Götze F.

Open Mathematics

2011

tom 9

nr 5

997-1050

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.

Limit theorems in free probability theory II

100%

Chistyakov G. , Götze F.

Open Mathematics

2008

tom 6

nr 1

87-117

Based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line ℝ+ and on the unit circle $$ \mathbb{T} $$ we prove analogs of limit theorems for nonidentically distributed random variables in classical Probability Theory.