The arithmetic of distributions in free probability theory

Chistyakov G, Götze F (2011)
Central European Journal of Mathematics 9(5): 997-1050.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
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.
Stichworte
Cauchy transforms; Free random variables; Free convolutions; Delphic; Khintchine's theorems; semigroups
Erscheinungsjahr
2011
Zeitschriftentitel
Central European Journal of Mathematics
Band
9
Ausgabe
5
Seite(n)
997-1050
ISSN
1895-1074
Page URI
https://pub.uni-bielefeld.de/record/2326392

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Chistyakov G, Götze F. The arithmetic of distributions in free probability theory. Central European Journal of Mathematics. 2011;9(5):997-1050.
Chistyakov, G., & Götze, F. (2011). The arithmetic of distributions in free probability theory. Central European Journal of Mathematics, 9(5), 997-1050. doi:10.2478/s11533-011-0049-4
Chistyakov, G., and Götze, F. (2011). The arithmetic of distributions in free probability theory. Central European Journal of Mathematics 9, 997-1050.
Chistyakov, G., & Götze, F., 2011. The arithmetic of distributions in free probability theory. Central European Journal of Mathematics, 9(5), p 997-1050.
G. Chistyakov and F. Götze, “The arithmetic of distributions in free probability theory”, Central European Journal of Mathematics, vol. 9, 2011, pp. 997-1050.
Chistyakov, G., Götze, F.: The arithmetic of distributions in free probability theory. Central European Journal of Mathematics. 9, 997-1050 (2011).
Chistyakov, Gennadiy, and Götze, Friedrich. “The arithmetic of distributions in free probability theory”. Central European Journal of Mathematics 9.5 (2011): 997-1050.