Cumulant-moment relation in free probability theory
2014 (English)In: Acta et Commentationes Universitatis Tartuensis de Mathematica, ISSN 1406-2283, E-ISSN 2228-4699, Vol. 18, no 2, 265-278 p.Article in journal (Refereed) Published
The goal of this paper is to present and prove a cumulant-moment recurrent relation formula in free probability theory. It is convenient tool to determine underlying compactly supported distribution function. The existing recurrent relations between these objects require the combinatorial understanding of the idea of non-crossing partitions, which has been considered by Speicher and Nica. Furthermore, some formulations are given with additional use of the Möbius function. The recursive result derived in this paper does not require introducing any of those concepts. Similarly like the non-recursive formulation of Mottelson our formula demands only summing over partitions of the set. The proof of non-recurrent result is given with use of Lagrange inversion formula, while in our proof the calculations of the Stieltjes transform of the underlying measure are essential.
Place, publisher, year, edition, pages
Tartu University Press, 2014. Vol. 18, no 2, 265-278 p.
R-transform, Free cumulants, Moments, Free probability, Non-commutative probability space, Stieltjes transform, Random matrices
Probability Theory and Statistics Other Mathematics
IdentifiersURN: urn:nbn:se:lnu:diva-58167DOI: 10.12697/ACUTM.2014.18.22OAI: oai:DiVA.org:lnu-58167DiVA: diva2:1047431