Marčenko-Pastur law for Tyler's M-estimator

This paper studies the limiting behavior of Tyler's M-estimator for the scatter matrix, in the regime that the number of samples n and their 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 x₁, . . ., x_n are identically and independently generated from the Gaussian distribution N(0,I), the operator norm of the difference between a properly scaled Tyler's M-estimator and ∑_i=1ⁿx_ix_i^⊤/n tends to zero. As a result, the spectral distribution of Tyler's M-estimator converges weakly to the Marčenko-Pastur distribution.