Some Limit Theorems on the Eigenvectors of Large Dimensional Sample Covariance Matrices

Let (vu) i, j = 1, 2 ,..., be i.i.d. standardized random variables. For each n, let V, = (v,,) i = 1, 2 ,..., n; j = 1, 2 ,..., s = s(n), where (n/s) -+ y > 0 as n + 03, and let M, = (l/s) V, Vi. Previous results [7,8] have shown the eigenvectors of M, to display behavior, for n large, similar to those of the corresponding Wishart matrix. A certain stochastic process X, on [0, 11, constructed from the eigenvectors of M,, is known to converge weakly, as n + co, on D[O, l] to Brownian bridge when a,, is N(0, 1), but it is not known whether this property holds for any other distribution. The present paper provides evidence that this property may hold in the non-Wishart case in the form of limit theorems on the convergence in distribution of random variables constructed from integrating analytic function w.r.t. X,(F,(x)), where F, is the empirical distribution function of the eigenvalues of M,. The theorems assume certain conditions on the moments of v,, including E(v:,) = 3, the latter being necessary for the theorems to hold. Q 1984 Academic eras, IIIC.