Let $S_n=\frac{ 1}{ n}X_nX_n^*$ where $X_n=\{ X_{ ij}\}$ is a $p\times n$ matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_n(\mathbf { t}_1,\mathbf { t}_2,\sigma)=\sqrt{ p}({ \mathbf { x}}_n(\mathbf { t}_1)^*(S_n+\sigma I)^{ -1}{ \mathbf { x}}_n(\mathbf { t}_2)-{ \mathbf { x}}_n(\mathbf { t}_1)^*{ \mathbf { x}}_n(\mathbf { t}_2)m_n(\sigma))$ in which $\sigma>0$ and $m_n(\sigma)=\int\frac{ dF_{ y_n}(x)}{ x+\sigma}$ where $F_{ y_n}(x)$ is the Mar\v{ c}enko--Pastur law with parameter $y_n=p/n$; which converges to a positive constant as $n\to\infty$, and ${ \mathbf { x}}_n(\mathbf { t}_1)$ and ${ \mathbf { x}}_n(\mathbf { t}_2)$ are unit vectors in ${ \Bbb{ C}}^p$, 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.
Categories: math.PR