Tracy-Widom limit for the largest eigenvalue of high-dimensional covariance matrices in elliptical distributions

Abstract

Let X be an M×N random matrix consisting of independent M-variate elliptically distributed column vectors x1,…,xN with general population covariance matrix Σ. In the literature, the quantity XX∗ is referred to as the sample covariance matrix after scaling, where X∗ is the transpose of X. In this article, we prove that the limiting behavior of the scaled largest eigenvalue of XX∗ is universal for a wide class of elliptical distributions, namely, the scaled largest eigenvalue converges weakly to the same limit regardless of the distributions that x1,…,xN follow as M,N→∞ with M∕N→ϕ0>0 if the weak fourth moment of the radius of x1 exists. In particular, via comparing the Green function with that of the sample covariance matrix of multivariate normally distributed data, we conclude that the limiting distribution of the scaled largest eigenvalue is the celebrated Tracy-Widom law.