Limiting Behavior Of Eigenvectors Of Large Dimensional Random Matrices

This thesis is concerned on finding the limiting behavior of eigenvectors oflarge sample covariance matrices.Based on the observation that the limiting spectral properties of large dimen-sional sample covariance matrix are asymptotically distribution free and the factthat the matrix of eigenvectors (eigenmatrix) of the Wishart matrix is Haar dis-tributed over the group of unitary matrices, it is conjectured that the behavior ofeigenmatrix of a large sample covariance matrix should asymptotically perform asHaar distributed under some moment conditions.The main work in this thesis involves two parts. In the first part (Chapter2),to investigate the limiting behavior of eigenvectors of a large sample covariance ma-trix and Wigner matrix, we define the eigenVector Empirical Spectral Distribution(VESD) with weights defined by eigenvectors and establish three types of conver-gence rates of the VESD when data dimension and sample size proportionallytend to infinity.In the second part (Chapter3), the limiting behavior of eigenvectors of samplecovariance matrices is further discussed. Using Bernstein polynomial approxima-tion and results obtained in Chapter2, we prove the central limit theorem for thelinear spectral statistics associated with the VESD, indexed by a set of functionswith continuous third order derivatives. This result provides us a strong evidenceto conjecture that the eigenmatrix of large sample covariance matrices is asymp-totically Haar distributed.