The arithmetic of distributions in free probability theory

作者：Chistyakov Gennadii P^*; Goetze Friedrich

来源：Central European Journal of Mathematics, 2011, 9(5): 997-1050.

DOI：10.2478/s11533-011-0049-4

• Free random variables
• Cauchy transforms
• Free convolutions
• Delphic semigroups
• Khintchine's theorems

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.

• 出版日期2011-10