## The Arithmetic of Distributions in Free Probability Theory (2005)

Citations: | 6 - 2 self |

### BibTeX

@TECHREPORT{Chistyakov05thearithmetic,

author = {G. P. Chistyakov and F. Götze},

title = { The Arithmetic of Distributions in Free Probability Theory},

institution = {},

year = {2005}

}

### OpenURL

### Abstract

We give a new 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 M equipped with the operation of free convolution and prove a Khintchine type theorem for factorization of elements of this semigroup. Any element of M contains either indecomposable (“prime”) factors or it belongs to a class, say I0, of distributions without indecomposable factors. In contrast to the classical convolution semigroup in the free additive and multiplicative convolution semigroups the class I0 consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in M.