The Arithmetic of Distributions in Free Probability Theory

We give an analytical 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 $\bold M$ equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of $\bold M$ 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 $\bold M$.Comment: 66 pages; latex; 5 figures; corrected version of proofs of Khintchine type theorems. For details see end of introductio

Concentration of empirical distribution functions with applications to non-i.i.d. models

The concentration of empirical measures is studied for dependent data, whose joint distribution satisfies Poincar\'{e}-type or logarithmic Sobolev inequalities. The general concentration results are then applied to spectral empirical distribution functions associated with high-dimensional random matrices.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ254 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm