Asymptotic properties of eigenmatrices of a large sample covariance matrix

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 $Yₙ(t₁,t₂,σ)=√p({x}ₙ(t₁)*(Sₙ+σ I)⁻¹{x}ₙ(t₂)-{x}ₙ(t₁)*{x}ₙ(t₂)mₙ(σ))$ 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 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.