Linear spectral statistics of eigenvectors of anisotropic sample covariance matrices

TL;DR

This paper investigates the asymptotic behavior of eigenvectors of anisotropic sample covariance matrices, establishing a central limit theorem for their spectral statistics and deriving explicit covariance formulas.

Contribution

It introduces a functional CLT for eigenvector spectral statistics of anisotropic covariance matrices, with explicit covariance expressions depending on the population covariance and vectors.

Findings

Convergence of spectral statistics to Gaussian processes on global and local scales

Explicit covariance formulas involving $\

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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.