Independences and partial R-transforms in bi-free probability
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.
- Publication:
-
Annales de L'Institut Henri Poincare Section (B) Probability and Statistics
- Pub Date:
- August 2016
- DOI:
- 10.1214/15-AIHP691
- arXiv:
- arXiv:1410.4265
- Bibcode:
- 2016AIHPB..52.1437S
- Keywords:
-
- Mathematics - Operator Algebras;
- Mathematics - Probability
- E-Print:
- Annales de l'Institut Henri Poincar\'e (B) Probabilit\'es et Statistiques 52 (2016), no. 3, 1437-1473