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 H\"older 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.