Linear spectral statistics of eigenvectors of anisotropic sample covariance matrices

Abstract

Consider sample covariance matrices of the form $Q:={\Sigma }^{1/2}X{X}^{\top }{\Sigma }^{1/2}$, where $X=(x_{ij})$ is an $n\times N$ random matrix whose entries are independent random variables with mean zero and variance $N^{-1}$, and $\Sigma$ is a deterministic positive-definite covariance matrix. We study the limiting behavior of the eigenvectors of $Q$ through the so-called eigenvector empirical spectral distribution $F_{\mathbf{v}}$, which is an alternative form of empirical spectral distribution with weights given by $|{\mathbf{v}}^{\top }{\xi }_k{|}^2$, where $\mathbf{v}$ is a deterministic unit vector and ${\xi }_k$ are the eigenvectors of $Q$. We prove a functional central limit theorem for the linear spectral statistics of $F_{\mathbf{v}}$, indexed by functions with Hölder continuous derivatives. We show that the linear spectral statistics converge to some Gaussian processes both on global scales of order 1 and on local scales that are much smaller than 1 but much larger than the typical eigenvalue spacing $N^{-1}$. Moreover, we give explicit expressions for the covariance functions of the Gaussian processes, where the exact dependence on $\Sigma$ and $\mathbf{v}$ is identified for the first time in the literature.

Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques (to appear)

References in corpus (5)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• High Dimensional Statistical Inference and Random Matrices
• The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices
• Spiked separable covariance matrices and principal components
• Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices

Cited by in corpus (1)

• Eigenvector overlaps in large sample covariance matrices and nonlinear shrinkage estimators