A constant regression characterization of a Marchenko-Pastur law

TL;DR

This paper characterizes the Marchenko-Pastur distribution using Lukacs type conditions in free probability, showing that certain constant conditional moments imply the distributions of involved variables are Marchenko-Pastur.

Contribution

It provides a new Lukacs type characterization of the Marchenko-Pastur law within free probability theory.

Findings

Conditional moments of order 1 and -1 are constant for the sum of free variables.

Such conditions imply the individual variables follow the Marchenko-Pastur distribution.

The result extends classical characterizations to free probability setting.

Abstract

Lukacs type characterization of Marchenko--Pastur distribution in free probability is studied here. We prove that for free $\mathbb{X}$ and $\mathbb{Y}$ when conditional moments of order $1$ and $-1$ of $(\mathbb{X+Y})^{-1/2}\mathbb{X}(\mathbb{X+Y})^{-1/2}$ given $\mathbb{X+Y}$ are constant then $\mathbb{X}$ and $\mathbb{Y}$ have Marchenko--Pastur distribution.