ABSTRACT
We define free independence by the vanishing of all mixed free cumulants. Free independence is a rich source of non-commutative variables and probability laws. It is also a crucial tool to explore relations between random matrices when their dimensions grow.
Existence of freely independent variables with specified distributions is guaranteed by a construction of free product of *-probability spaces, analogous to the construction of product probability spaces. Details of this construction are given in the last Chapter.
Free binomial, circular and elliptic variables are defined. The semi-circular family is defined and it is the non-commutative analog of the multivariate Gaussian vector. The free analog of Isserlis' formula is established. Free additive convolution of compactly supported probability laws is briefly introduced.
A crucial tool in the combinatorics of non-crossing partitions is the Kreweras complementation map. We provide some of its basic properties. We use it to establish relations between moments and free cumulants of polynomials of free variables and also to identify the distribution of variables from given moments and free cumulants. Compound free Poisson variable, an analog of the classical compound Poisson random variable is introduced.
