Asymptotic properties of eigenmatrices of a large sample covariance matrix

Let $S_n=\frac{1}{n}{X}_n{X}_n^{\ast }$ where $X_n=\{X_{ij}\}$ is a $p\times n$ matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_n(\mathbf {t}_1,\mathbf {t}_2,\sigma)=\sqrt{p}({\mathbf {x}}_n(\mathbf {t}_1)^(S_n+\sigma I)^{-1}{\mathbf {x}}_n(\mathbf {t}_2)-{\mathbf {x}}_n(\mathbf {t}_1)^{\mathbf {x}}n(\mathbf {t}2)m_n(\sigma))$inwhich$\sigma>0$and$m_n(\sigma)=\int\frac{dF{y_n}(x)}{x+\sigma}$where$F{y_n}(x)$istheMar\check{c}enko--Pasturlawwithparameter$y_n=p/n$; which converges to a positive constant as $n\to \mathrm{\infty }$, and ${\mathbf {x}}_n(\mathbf {t}_1)$and${\mathbf {x}}_n(\mathbf {t}_2)$areunitvectorsin${\Bbb{C}}^p$, having indices ${\mathbf{t}}_1$ and ${\mathbf{t}}_2$, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence $Y_n({\mathbf{t}}_1,{\mathbf{t}}_2,\sigma )$ 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.