Two dimensional Meixner random vectors of class ${\mathcal M}_L$

TL;DR

This paper develops a framework for decomposing non-commutative random variables into annihilation, preservation, and creation parts, and classifies two-variable systems with Lie algebra structures as equivalent to independent Meixner variables.

Contribution

It introduces a unique joint decomposition for non-commutative variables and characterizes systems with Lie algebra structures as equivalent to Meixner random variables.

Findings

Decomposition of non-commutative variables is essentially unique.

Lie algebra structures correspond to systems of independent Meixner variables.

Commuting variables are equivalent to classical Meixner variables.

Abstract

The paper is divided into two parts. In the first part we lay down the foundation for defining the joint annihilation-preservation-creation decomposition of a finite family of, not necessarily commutative random variables, and show that this decomposition is essentially unique. In the second part we show that any two, not necessarily commutative, random variables $X$ and $Y$, for which the vector space spanned by their annihilation, preservation, and creation operators equipped with the bracket given by the commutator, forms a Lie algebra, are equivalent, up to an invertible linear transformation to two independent Meixner random variables with mixed preservation operators. In particular if $X$ and $Y$ commute, then they are equivalent, up to an invertible linear transformation to two independent classic Meixner random variables. To show this we start with a small technical condition called "non-degeneracy".