Limiting behavior of eigenvectors of large dimensional random matrices

Abstract

The main work in the thesis is to investigate the limiting behavior of eigenvectors of large sample covariance matrices. Based on the observation that the limiting spectral properties of large dimensional sample covariance matrix are asymptotically distribution free and the fact that the matrix of eigenvectors (eigenmatrix) of the Wishart matrix is uniformly distributed (i.e., Haar distributed) over the group of unitary matrices, it is conjectured that the behavior of eigenmatrix of a large sample covariance matrix should asymptotically perform as Haar distributed under some moment conditions. To investigate the limiting behavior of eigenvectors of a large sample covariance matrix, we define the eigenvector empirical spectral distribution (VESD) with weights defined by eigenvectors and establish the convergence rates and central limit theorem for the VESD.