Some limit theorems on the eigenvectors of large dimensional sample covariance matrices

Let {vij} i,j = 1, 2,..., be i.i.d. standardized random variables. For each n, let Vn = (vij) I = 1, 2,..., n; J = 1, 2,..., S = s(n), where (n/s) --> y > 0 as n --> [infinity], and let Mn = (1/s)VnVnT. Previous results [7, 8] have shown the eigenvectors of Mn to display behavior, for n large, similar to those of the corresponding Wishart matrix. A certain stochastic process Xn on [0, 1], constructed from the eigenvectors of Mn, is known to converge weakly, as n --> [infinity], on D[0, 1] to Brownian bridge when v11 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. Xn(Fn(x)), where Fn is the empirical distribution function of the eigenvalues of Mn. The theorems assume certain conditions on the moments of v11 including E(v114) = 3, the latter being necessary for the theorems to hold.