# Asymptotic properties of eigenmatrices of a large sample covariance
matrix

Research paper by **Z. D. Bai, H. X. Liu, W. K. Wong**

Indexed on: **30 Dec '11**Published on: **30 Dec '11**Published in: **Mathematics - Probability**

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#### Abstract

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.