The arithmetic of distributions in free probability theory

• Autorzy: Gennadii Chistyakov , Friedrich Götze
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Free random variables; Cauchy transforms; Free convolutions; Delphic semigroups; Khintchine’s theorems

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 5
• Strony: 997-1050

Daty

wydano

2011-10-01

online

2011-07-26

Twórcy

autor

Gennadii Chistyakov

autor

Friedrich Götze

• Universität Bielefeld

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Identyfikatory

DOI

10.2478/s11533-011-0049-4