On the eigenvectors of large dimensional sample covariance matrices

Abstract

Let { v ij }, i, j = 1,2, …, be i.i.d. random variables, and for each n let M n = ( 1 s )V nV n T , where V n = ( v ij ), i = 1, 2, …, n, j = 1, 2, …, s = s( n), and n s → y > 0 as n → ∞. Necessary and sufficient conditions are given to establish the convergence in distribution of certain random variables defined by M n . When E( v 11 4) < ∞ these variables play an important role toward understanding the behavior of the eigenvectors of this class of sample covariance matrices for n large.