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

Document Type

Book Chapter

Publication Title

Indian Statistical Institute Series

Abstract

This is a selective review on the behavior of the high-dimensional sample covariance matrix, S=n-1XX∗, 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 τ and Spearman’s ρ matrices, and the autocovariance matrices in one and high dimensions. Finally, we present some statistical applications.

First Page

11

Last Page

67

DOI

10.1007/978-981-99-9994-1_2

Publication Date

1-1-2024

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