Marchenko-Pastur Law for Tyler's M-estimator

Abstract: 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 $x_1, \ldots, x_n$ are identically and independently generated from the Gaussian distribution $\mathcal{N}(0, I)$, the operator norm of the difference between a properly scaled Tyler's M-estimator and $\sum_{i=1}^n x_i x_i^\top/n$ tends to zero. As a result, the spectral distribution of Tyler's M-estimator converges weakly to the Mar\v{c}enko-Pastur distribution.