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Eta-diagonal distributions and infinite divisibility for R-diagonals

Eta-diagonal distributions and infinite divisibility for R-diagonals

Authors :
Michael Noyes
Hari Bercovici
Kamil Szpojankowski
Alexandru Nica
Source :
Ann. Inst. H. Poincaré Probab. Statist. 54, no. 2 (2018), 907-937
Publication Year :
2016
Publisher :
arXiv, 2016.

Abstract

The class of R-diagonal *-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the relation between free probability and the parallel (and simpler) world of Boolean probability. It is natural to introduce the concept of an eta-diagonal distribution that is the Boolean counterpart of an R-diagonal distribution. We establish a number of properties of eta-diagonal distributions, then we examine the canonical bijection relating eta-diagonal distributions to infinitely divisible R-diagonal ones. The overall result is a parametrization of an arbitrary $\boxplus$-infinitely divisible R-diagonal distribution that can arise in a C*-probability space, by a pair of compactly supported Borel probability measures on $[ 0, \infty )$. Among the applications of this parametrization, we prove that the set of $\boxplus$-infinitely divisible R-diagonal distributions is closed under the operation $\boxtimes$ of free multiplicative convolution.<br />Comment: 33 pages

Details

Database :
OpenAIRE
Journal :
Ann. Inst. H. Poincaré Probab. Statist. 54, no. 2 (2018), 907-937
Accession number :
edsair.doi.dedup.....d56fd6d983a77ca63b72b182f0b33769
Full Text :
https://doi.org/10.48550/arxiv.1608.03515