A constant regression characterization of the Marchenko– Pastur law

Abstrakt

In this paper, Lukacs type characterization of Marchenko–Pastur distribution in free probability is studied. We prove that for free X and Y, if conditional moments of order 1 and −1 of
X + Y^−1/2XX + Y^−1/2 given X + Y are constant, then X and Y follow the Marchenko–Pastur distribution.