On asymptotics of eigenvectors of large sample covariance matrix

Abstract

Let {X_ij}, 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 T_n^1/2 is the square root of a nonnegative definite matrix T_n and X_n is the n×N matrix of the upper-left corner of the double array. The matrix A_n can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix T_n, or as a multivariate F matrix if T_n is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of A_n, 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 {X_ij} and T_n are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of A_n are proved to have Gaussian limits, which suggests that the eigenvector matrix of A_n is nearly Haar distributed when T_n is a multiple of the identity matrix, an easy consequence for a Wishart matrix.