Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Independences and partial $R$-transforms in bi-free probability

Paul Skoufranis

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Abstract

In this paper, we examine how various notions of independence in non-commutative probability theory arise in bi-free probability. We exhibit how Boolean and monotone independence occur from bi-free pairs of faces and establish a Kac/Loeve Theorem for bi-free independence. In addition, we prove that bi-freeness is preserved under tensoring with matrices. Finally, via combinatorial arguments, we construct partial $R$-transforms in two settings relating the moments and cumulants of a left–right pair of operators.

Résumé

Dans cet article, nous examinons comment diverses notions d’indépendance en théorie des probabilités non commutatives se traduisent en probabilités bi-libres. Nous montrons comment l’indépendance booléenne et monotone se produisent à partir de paires de faces bi-libres, et établissons un théorème de Kac/Loève pour l’indépendance bi-libre. En outre, nous prouvons que l’indépendance bi-libre est préservée par tensorisation avec des matrices. Enfin, par des arguments combinatoires, nous construisons deux types de $R$-transformations partielles, reliant les moments et les cumulants d’une paire gauche-droite des opérateurs.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1437-1473.

Dates
Received: 4 December 2014
Revised: 10 May 2015
Accepted: 31 May 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723526

Digital Object Identifier
doi:10.1214/15-AIHP691

Mathematical Reviews number (MathSciNet)
MR3531715

Zentralblatt MATH identifier
1364.46058

Subjects
Primary: 46L54: Free probability and free operator algebras 46L53: Noncommutative probability and statistics

Keywords
Bi-free probability Free independence Boolean independence Monotone independence Bi-free independence over matrices Partial $R$-transforms

Citation

Skoufranis, Paul. Independences and partial $R$-transforms in bi-free probability. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1437--1473. doi:10.1214/15-AIHP691. https://projecteuclid.org/euclid.aihp/1469723526


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