Characterizations of some free random variables by properties of conditional moments of third degree polynomials

TL;DR

This paper characterizes certain free random variables using properties of third-degree polynomial conditional moments, advancing understanding of free Meixner distributions and their relation to free Levy processes.

Contribution

It introduces a novel characterization of free Meixner distributions via third-degree polynomial conditional moments, extending to q-Gaussian variables.

Findings

Free Meixner distributions characterized by third-degree polynomial moments

Implications for describing free Levy processes

Extension of results to q-Gaussian variables

Abstract

We investigate Laha-Lukacs properties of noncommutative random variables (processes). We prove that some families of free Meixner distributions can be characterized by the conditional moments of polynomial functions of degree 3. We also show that this fact has consequences in describing some free Levy processes. The proof relies on a combinatorial identity. At the end of this paper we show that this result can be extended to a q-Gausian variable.