Asymptotic Properties of Eigenmatrices of a Large Sample Covariance Matrix By

Let Sn = 1 nXnX∗ n where Xn = {Xij } is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Yn(t1, t2, σ ) = √ p(xn(t1)∗(Sn + σI)−1xn(t2)− xn(t1)∗xn(t2)mn(σ )) in which σ > 0 and mn(σ)= ∫ dFyn (x) x+σ where Fyn(x) is the Marčenko–Pastur law with parameter yn = p/n; which converges to a positive constant as n→∞, and xn(t1) and xn(t2) are unit vectors in Cp , having indices t1 and t2, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Yn(t1, t2, σ ) converges weakly to a (2m+ 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of Sn is asymptotically close to that of a Haar-distributed unitary matrix.