Asymptotic properties of eigenmatrices of a large sample covariance matrix

Abstract

Let S_n = 1/n X_nX_n* where X_n = {X_ij} 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 m_n(σ) = ∫d F_yn(x)/(x + σ) where F_yn(x) is the Marčenko–Pastur law with parameter y_n = p/n; which converges to a positive constant as n → ∞, and x_n(t₁) and x_n(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 Y_n(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 S_n is asymptotically close to that of a Haar-distributed unitary matrix.