Asymptotic properties of eigenmatrices of a large sample covariance matrix

Abstract

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 $Y_n({\mathbf{t}}_1,{\mathbf{t}}_2,\sigma )=\sqrt{p}({\mathbf{x}}_n({\mathbf{t}}_1{)}^{\ast }(S_n+\sigma I{)}^{-1}{\mathbf{x}}_n({\mathbf{t}}_2)-{\mathbf{x}}_n({\mathbf{t}}_1{)}^{\ast }{\mathbf{x}}_n({\mathbf{t}}_2)m_n(\sigma ))$ in which $\sigma >0$ and $m_n(\sigma )=\int \frac{d{F}_{y_n}(x)}{x+\sigma }$ where $F_{y_n}(x)$ is the Marčenko--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 ${\mathbb{C}}^p$, 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.

arXiv commentPublished in at http://dx.doi.org/10.1214/10-AAP748 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• On asymptotics of eigenvectors of large sample covariance matrix

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