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

Fevrier, Maxime; Mastnak, Mitja; Nica, Alexandru; Szpojankowski, Kamil

doi:10.1090/tran/8122

Using Boolean cumulants to study multiplication and anti-commutators of free random variables
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by Maxime Fevrier, Mitja Mastnak, Alexandru Nica and Kamil Szpojankowski PDF

Trans. Amer. Math. Soc. 373 (2020), 7167-7205 Request permission

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.

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