Conditionally Bi-Free Independence for Pairs of Algebras

TL;DR

This paper introduces conditionally bi-free independence for pairs of algebras, develops associated cumulants, and establishes limit theorems including a bi-free Lévy-Hinčin formula, advancing non-commutative probability theory.

Contribution

It defines conditionally bi-free independence, introduces conditional cumulants, and derives a bi-free Lévy-Hinčin formula, extending free probability concepts.

Findings

Conditional bi-free independence is equivalent to mixed cumulants.

A conditionally bi-free partial R-transform is constructed.

A bi-free Lévy-Hinčin formula for planar measures is established.

Abstract

In this paper, the notion of conditionally bi-free independence for pairs of algebras is introduced. The notion of conditional $(\ell, r)$-cumulants are introduced and it is demonstrated that conditionally bi-free independence is equivalent to mixed cumulants. Furthermore, limit theorems for the additive conditionally bi-free convolution are studied using both combinatorial and analytic techniques. In particular, a conditionally bi-free partial $\mathcal{R}$-transform is constructed and a conditionally bi-free analogue of the L\'{e}vy-Hin\v{c}in formula for planar Borel probability measures is derived.