We will present a result which relates Boolean cumulants and free random variables.
First we will provide a characterization of freeness in terms of Boolean cumulants. Our next goal will be to present that this characterisation is useful. We will show that for some natural problems in free probability, combinatorics of Boolean cumulants of free random variables is easier than combinatorics of free cumulants. In particular we will discuss the combinatorics of free multiplicative convolution of $*$--distributions in terms of Boolean cumulants. We will present an application of Boolean cumulants approach to subordination results in free probability. Among applications of we will describe the distribution of $X+f(X)Yf(X)$ for free random variables $X,Y$ as well as the distribution of anticommutator $XY+YX$.
Talk is based on two joint projects:
1) with M. Fevrier (Paris, France), M. Mastnak (Halifax, Canada) and A. Nica (Waterloo, Canada),
2) with F. Lehner (Graz, Austria).