Spectrum of High-Dimensional Sample Covariance and Related Matrices: A Selective Review

This is a selective review on the behavior of the high-dimensional sample covariance matrix, \(S=n^{-1}XX^{*}\) , the most important random matrix in high-dimensional statistics, and some related matrices. The asymptotic distribution of the empirical spectrum of (centered and scaled) S is either the semi-circular law or the Marčenko–Pastur law, depending on how the dimensions n and p grow. From a non-commutative probability point of view, independent copies of S converge algebraically to freely independent semi-circular or compound free Poisson variables. We also discuss the asymptotic normality of the linear spectral statistics, and the convergence of the maximum eigenvalue to a Tracy–Widom law. Some related matrices are also discussed. These include the separable, generalized, and cross-covariance matrices, Kendall’s \(\tau \) and Spearman’s \(\rho \) matrices, and the autocovariance matrices in one and high dimensions. Finally, we present some statistical applications.