Limiting Behavior of Eigenvectors of Large Wigner Matrices

A new form of empirical spectral distribution of a Wigner matrix W n with weights specified by the eigenvectors is defined and it is then shown to converge with probability one to the semicircular law. Moreover, central limit theorem for linear spectral statistics defined by the eigenvectors and eigenvalues is also established under some moment conditions, which suggests that the eigenvector matrix of W n is close to being Haar distributed.