Comments on a result of Yin, Bai, and Krishnaiah for large dimensional multivariate F matrices

Let {v_ij} i,j = 1, 2,…, be i.i.d. standardized random variables. For each n, let V_n = (v_ij) I = 1, 2,…, n; J = 1, 2,…, S = s(n), where (n/s) → y > 0 as n → ∞, and let M_n = (1/s)V_nV_n^T. Previous results 7, 8] have shown the eigenvectors of M_n to display behavior, for n large, similar to those of the corresponding Wishart matrix. A certain stochastic process X_n on 0, 1], constructed from the eigenvectors of M_n, is known to converge weakly, as n → ∞, on D0, 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. X_n(F_n(x)), where F_n is the empirical distribution function of the eigenvalues of M_n. The theorems assume certain conditions on the moments of v₁₁ including E(v₁₁⁴) = 3, the latter being necessary for the theorems to hold.