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

Abstract: Let $X$ be an $M\times N$ random matrix consisting of independent $M$-variate elliptically distributed column vectors $\mathbf{x}{1},\dots,\mathbf{x}{N}$ with general population covariance matrix $\Sigma$. 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 $\mathbf{x}{1},\dots,\mathbf{x}{N}$ follow as $M,N\to\infty$ with $M/N\to\phi_0>0$ if the weak fourth moment of the radius of $\mathbf{x}_{1}$ 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.