On asymptotics of eigenvectors of large sample covariance matrix

Abstract

Let {Xij}, i, j=…, be a double array of i.i.d. complex random variables with EX11=0,E|X11|2=1 and E|X11|4<∞, and let $A_{n}=\frac{1}{N}T_{n}^{{1}/{2}}X_{n}X_{n}^{*}T_{n}^{{1}/{2}}$, where Tn1/2 is the square root of a nonnegative definite matrix Tn and Xn is the n×N matrix of the upper-left corner of the double array. The matrix An can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix Tn, or as a multivariate F matrix if Tn is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of An, a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if {Xij} and Tn are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of An are proved to have Gaussian limits, which suggests that the eigenvector matrix of An is nearly Haar distributed when Tn is a multiple of the identity matrix, an easy consequence for a Wishart matrix.