  The eigenvector Empirical Spectral Distribution (VESD) is adopted to
investigate the limiting behavior of eigenvectors and eigenvalues of covariance
matrices. In this paper, we shall show that the Kolmogorov distance between the
expected VESD of sample covariance matrix and the Mar\v{c}enko-Pastur
distribution function is of order $O(N^{-1/2})$. Given that data dimension $n$
to sample size $N$ ratio is bounded between 0 and 1, this convergence rate is
established under finite 10th moment condition of the underlying distribution.
It is also shown that, for any fixed $\eta>0$, the convergence rates of VESD
are $O(N^{-1/4})$ in probability and $O(N^{-1/4+\eta})$ almost surely,
requiring finite 8th moment of the underlying distribution.
