Linear spectral statistics of eigenvectors of anisotropic sample covariance matrices

Consider sample covariance matrices of the form $Q:=\Sigma^{1/2} X X^* \Sigma^{1/2}$, 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 matrix. We study the limiting behavior of the eigenvectors of $Q$ through the so-called eigenvector empirical spectral distribution (VESD) $F_{\mathbf u}$, which is an alternate form of empirical spectral distribution with weights given by $|\mathbf u^\top \xi_k|^2$, where $\mathbf u$ is any deterministic unit vector and $\xi_k$ are the eigenvectors of $Q$. We prove a functional central limit theorem for the linear spectral statistics of $F_{\mathbf u}$, indexed by functions with H{\"o}lder continuous derivatives. We show that the linear spectral statistics converge to universal Gaussian processes both on global scales of order 1, and on local scales that are much smaller than 1 and much larger than the typical eigenvalues spacing $N^{-1}$. Moreover, we give explicit expressions for the means and covariance functions of the Gaussian processes, where the exact dependence on $\Sigma$ and $\mathbf u$ allows for more flexibility in the applications of VESD in statistical estimations of sample covariance matrices.