## 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.## References

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## Additional Information

**Maxime Fevrier**- Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
- MR Author ID: 893902
- Email: maxime.fevrier@u-psud.fr
**Mitja Mastnak**- Affiliation: Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia B3H 3C3, Canada
- MR Author ID: 695207
- Email: mmastnak@cs.smu.ca
**Alexandru Nica**- Affiliation: Department of Pure Mathematics, University of Waterloo, Ontario, Canada
- Email: anica@uwaterloo.ca
**Kamil Szpojankowski**- Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
- MR Author ID: 1041201
- Email: k.szpojankowski@mini.pw.edu.pl
- Received by editor(s): September 4, 2019
- Received by editor(s) in revised form: January 21, 2020
- Published electronically: July 28, 2020
- Additional Notes: The research of the second and third authors was supported by a Discovery Grant from NSERC, Canada.

The research of the fourth author was partially suported by NCN grant 2016/23/D/ST1/01077. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 7167-7205 - MSC (2010): Primary 46L54; Secondary 46L53, 05A18, 60C05, 60B20
- DOI: https://doi.org/10.1090/tran/8122
- MathSciNet review: 4155204