A constant regression characterization of a Marchenko-Pastur law

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 $(\mathbb{X}+\mathbb{Y}{)}^{-1/2}\mathbb{X}(\mathbb{X}+\mathbb{Y}{)}^{-1/2}$ given $\mathbb{X}+\mathbb{Y}$ are constant then $\mathbb{X}$ and $\mathbb{Y}$ have Marchenko--Pastur distribution.

References in corpus (1)

• Dual Lukacs regressions of negative orders for non-commutative variables

Cited by in corpus (1)

• Free infinite divisibility for powers of random variables