BMT Independence

TL;DR

This paper introduces BMT independence, a new framework combining boolean, monotone, and tensor independence, with limit theorems and operator models, advancing non-commutative probability theory.

Contribution

It generalizes existing independence notions by encoding pair-wise relations via directed graphs and provides explicit constructions and limit theorems for BMT independent variables.

Findings

Established BMT independence as a unifying framework.

Derived Central and Poisson-Type Limit Theorems for BMT independence.

Provided explicit operator models for BMT independent variables.

Abstract

We introduce the notion of BMT independence, allowing us to take arbitrary mixtures of boolean, monotone, and tensor independence and generalizing the notion of BM independence of Wysoczanski. Pair-wise independence relations are encoded through a directed graph, which in turn determines the way mixed moments must be computed. Corresponding Central and Poisson-Type Limit Theorems are provided along with an explicit construction to realize BMT independent random variables as bounded operators on certain Hilbert space.