1--Meixner random vectors

TL;DR

This paper introduces the concept of $d$-dimensional $n$-Meixner random vectors, derives PDEs for their Laplace transforms, and characterizes all non-degenerate three-dimensional cases, highlighting their unique non-reducibility.

Contribution

It defines $n$-Meixner random vectors using semi-quantum operators and provides a complete characterization of three-dimensional cases, revealing new structural properties.

Findings

Derived PDE system for Laplace transforms of $1$-Meixner vectors.

Established necessary conditions for PDE integrability.

Characterized all non-degenerate three-dimensional $1$-Meixner vectors.

Abstract

A definition of $d$--dimensional $n$--Meixner random vectors is given first. This definition involves the commutators of their semi--quantum operators. After that we will focus on the $1$-Meixner random vectors, and derive a system of $d$ partial differential equations satisfied by their Laplace transform. We provide a set of necessary conditions for this system to be integrable. We use these conditions to give a complete characterization of all non--degenerate three--dimensional $1$--Meixner random vectors. It must be mentioned that the three--dimensional case produces the first example in which the components of a $1$--Meixner random vector cannot be reduced, via an injective linear transformation, to three independent classic Meixner random variables.