Our main concern in this lecture will be the understanding and effective description of the sum of freely independent random variables. How can we calculate the distribution of a + b if a and b are free and if we know the distribution of a and the distribution of b. Of particular interest is the case of selfadjoint random variables x and y in a C*-probability space. In this case their distributions can be identified with probability measures on ℝ and thus taking the sum of free random variables gives rise to a binary operation on probability measures on ℝ. We will call this operation “free convolution,” in analogy with the usual concept of convolution of probability measures which corresponds to taking the sum of classically independent random variables. Our combinatorial approach to free probability theory, resting on the notion of free cumulants, will give us very easy access to the main results of Voiculescu on this free convolution via the so-called “R-transform.”

Free convolution

Definition 12.1. Let μ and ν be probability measures on ℝ with compact support. Let x and y be selfadjoint random variables in some C*-probability space such that x has distribution μ, y has distribution ν, and such that x and y are freely independent. Then the distribution of the sum x + y is called the free convolution of μ and μ and is denoted by μ ⊞ ν.