  In this paper, we adopt the eigenvector empirical spectral distribution
(VESD) to investigate the limiting behavior of eigenvectors of a large
dimensional Wigner matrix W_n. In particular, we derive the optimal bound for
the rate of convergence of the expected VESD of W_n to the semicircle law,
which is of order O(n^{-1/2}) under the assumption of having finite 10th
moment. We further show that the convergence rates in probability and almost
surely of the VESD are O(n^{-1/4}) and O(n^{-1/6}), respectively, under finite
8th moment condition. Numerical studies demonstrate that the convergence rate
does not depend on the choice of unit vector involved in the VESD function, and
the best possible bound for the rate of convergence of the VESD is of order
O(n^{-1/2}).
