Asymptotic properties of eigenmatrices of a large sample covariance matrix

Abstract

Let Sn = 1/n XnXn* where Xn = {Xij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_{n}(\mathbf{t}_{1},\mathbf{t}_{2},\sigma)=\sqrt{p}({\mathbf{x}}_{n}(\mathbf{t}_{1})^{*}(S_{n}+\sigma I)^{-1}{\mathbf{x}}_{n}(\mathbf{t}_{2})-{\mathbf{x}}_{n}(\mathbf{t}_{1})^{*}{\mathbf{x}}_{n}(\mathbf{t}_{2})m_{n}(\sigma))$ in which σ > 0 and mn(σ) = ∫d Fyn(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 ${\mathbb{C}}^{p}$, 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.