  The eigenvector empirical spectral distribution (VESD) is a useful tool in
studying the limiting behavior of eigenvalues and eigenvectors of covariance
matrices. In this paper, we study the convergence rate of the VESD of sample
covariance 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 entries
are independent random variables with mean zero and variance $N^{-1}$, and
$\Sigma$ is a deterministic positive-definite matrix. We prove that the
Kolmogorov distance between the expected VESD and the deformed MP distribution
is bounded by $N^{-1+\epsilon}$ for any fixed $\epsilon>0$, provided that the
entries $\sqrt{N}x_{ij}$ have uniformly bounded 6th moments and $|N/M-1|\ge
\tau$ for some constant $\tau>0$. This result improves the previous one
obtained 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, the
convergence rate of the VESD is $O(N^{-1/2+\epsilon})$ almost surely for any
fixed $\epsilon>0$, which improves the previous bound $N^{-1/4+\epsilon}$ in
\cite{XYZ2013}.
