Marchenko-Pastur Law for Tyler's and Maronna's M-estimators

This paper studies the limiting behavior of Tyler's and Maronna's M- estimators, in the regime that the number of samples n and the dimension p both go to infinity, and p/n converges to a constant y with 0 < y < 1. We prove that when the data samples are identically and independently generated from the Gaussian distribution N(0,I), the difference between the sample covariance matrix and a scaled version of Tyler's M-estimator or Maronna's M-estimator tends to zero in spectral norm, and the empirical spectral densities of both estimators converge to the Marchenko-Pastur distribution. We also extend this result to elliptical-distributed data samples for Tyler's M-estimator and non-isotropic Gaussian data samples for Maronna's M-estimator.