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 $\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ 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 Mar\v{c}enko-Pastur distribution. We also prove that when the data samples are generated from an elliptical distribution, the limiting distribution of Tyler's M-estimator converges to a Mar\v{c}enko-Pastur-Type distribution.