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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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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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:
  • Mitja Mastnak
  • Affiliation: Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia B3H 3C3, Canada
  • MR Author ID: 695207
  • Email:
  • Alexandru Nica
  • Affiliation: Department of Pure Mathematics, University of Waterloo, Ontario, Canada
  • Email:
  • Kamil Szpojankowski
  • Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
  • MR Author ID: 1041201
  • Email:
  • 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:
  • MathSciNet review: 4155204