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The eigenvector empirical spectral distribution (VESD) is a useful tool instudying the limiting behavior of eigenvalues and eigenvectors of covariancematrices. In this paper, we study the convergence rate of the VESD of samplecovariance matrices to the deformed Mar\v{c}enko-Pastur (MP) distribution.Consider sample covariance matrices of the form $\Sigma^{1/2} X X^*\Sigma^{1/2}$, where $X=(x_{ij})$ is an $M\times N$ random matrix whose entriesare independent random variables with mean zero and variance $N^{-1}$, and$\Sigma$ is a deterministic positive-definite matrix. We prove that theKolmogorov distance between the expected VESD and the deformed MP distributionis bounded by $N^{-1+\epsilon}$ for any fixed $\epsilon>0$, provided that theentries $\sqrt{N}x_{ij}$ have uniformly bounded 6th moments and $|N/M-1|\ge\tau$ for some constant $\tau>0$. This result improves the previous oneobtained in \cite{XYZ2013}, which gave the convergence rate $O(N^{-1/2})$assuming $i.i.d.$ $X$ entries, bounded 10th moment, $\Sigma=I$ and $M<N$.Moreover, we also prove that under the finite $8$th moment assumption, theconvergence rate of the VESD is $O(N^{-1/2+\epsilon})$ almost surely for anyfixed $\epsilon>0$, which improves the previous bound $N^{-1/4+\epsilon}$ in\cite{XYZ2013}.

Convergence of eigenvector empirical spectral distribution of sample covariance matrices