Using Boolean cumulants to study multiplication and anti-commutators of free random variables

Maxime Fevrier , Mitja Mastnak , Alexandru Nica , Kamil Szpojankowski

Abstract

We study how Boolean cumulants can be used in order to address operations with freely independent random variables, particularly in connection to the $ *$-distribution of the product of two selfadjoint freely independent random variables, and in connection to the distribution of the anti-commutator of such random variables. A key instrument in our considerations is a new combinatorial object, coloured noncrossing partitions with a structural property which we call vertical no-repeat property. As a byproduct, we obtain several results concerning enumeration of some special sets of noncrossing partitions.