Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Eta-diagonal distributions and infinite divisibility for R-diagonals

Hari Bercovici, Alexandru Nica, Michael Noyes, and Kamil Szpojankowski

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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.


Dans le cadre des probabilités libres, la classe des $*$-distributions $R$-diagonales est assez bien comprise. Dans cette classe, nous nous intéressons au concept d’infinie divisibilité par rapport à l’opération de convolution additive libre $\boxplus$. Nous exploitons la relation entre probabilités libres et le monde parallèle (et plus simple) des probabilités booléennes. Il est naturel d’introduire le concept de distributions $\eta$-diagonales, qui sont la contrepartie booléenne des distributions $R$-diagonales. Nous établissons un certain nombre de propriétés des distributions $\eta$-diagonales, avant d’examiner la bijection canonique reliant les distributions $\eta$-diagonales aux distributions $R$-diagonales indéfiniment divisibles. Le résultat principal est la paramétrisation par une paire de mesures boreliennes sur $[0,\infty)$ à support compact de toutes les lois $R$-diagonales $\boxplus$-indéfiniment divisibles pouvant apparaître dans un $C^{*}$-espace de probabilités. Parmi les applications de cette paramétrisation, nous montrons que l’ensemble des distributions $R$-diagonales $\boxplus$-indéfiniment divisibles est fermé sous l’opération $\boxtimes$ de convolution multiplicative libre.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 907-937.

Received: 29 August 2016
Revised: 13 February 2017
Accepted: 7 March 2017
First available in Project Euclid: 25 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L54: Free probability and free operator algebras
Secondary: 46L53: Noncommutative probability and statistics 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms

R-diagonal distribution $\eta$-diagonal distribution Free additive convolution Infinite divisibility


Bercovici, Hari; Nica, Alexandru; Noyes, Michael; Szpojankowski, Kamil. Eta-diagonal distributions and infinite divisibility for R-diagonals. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 907--937. doi:10.1214/17-AIHP826.

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