A three-state independence in non-commutative probability

TL;DR

This paper introduces a new independence concept called $$-freeness in non-commutative probability, unifying several existing independences and deriving a central limit theorem with Kesten distributions.

Contribution

It defines $$-freeness, unifies multiple independence types, and extends the central limit theorem to this new framework.

Findings

$$-freeness unifies free, monotone, antimonotone, and Boolean independences.

The associative law is preserved across these independences.

The central limit theorem yields triplets of Kesten distributions.

Abstract

We define a new independence in non-commutative probability, called $\alpha$-freeness, with respect to a triplet of states. This concept unifies several independences in non-commutative probability, in particular, free, monotone, antimonotone and Boolean ones as well as conditionally free, conditionally monotone and conditionally antimonotone independences. Moreover, the associative law of $\alpha$-freeness is transferred to the other independences. As a consequence, $\alpha$-free cumulants unify the cumulants for free, monotone, antimonotone and Boolean independences. The central limit theorem for $\alpha$-freeness is computed. The limit distribution turns out to be a triplet of the Kesten distributions.