The Annals of Probability

Weak Convergence of Random Functions Defined by The Eigenvectors of Sample Covariance Matrices

Jack W. Silverstein

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Abstract

Let $\{v_{ij}\}, i, j = 1, 2, \ldots,$ be i.i.d. symmetric random variables with $\mathbb{E}(\nu^4_{11}) < \infty$, and for each $n$ let $M_n = (1/s)V_n V^T_n$, where $V_n = (v_{ij}), i = 1, 2, \ldots, n, j = 1, 2, \ldots, s = s(n)$ and $n/s \rightarrow y > 0$ as $n \rightarrow \infty$. Denote by $O_n \Lambda_n O^T_n$ the spectral decomposition of $M_n$. Define $X \in D\lbrack 0, 1 \rbrack$ by $X_n(t) = \sqrt{n/2} \sum^{\lbrack nt \rbrack}_{i = 1}(y^2_i - 1/n)$ where $(y_1, y_2, \ldots, y_n)^T = O^T(\pm 1/\sqrt{n}, \pm 1/ \sqrt{n}, \ldots, \pm 1/\sqrt{n})^T$. It is shown that $X_n \rightarrow_\mathscr{D} W^0$ as $n \rightarrow \infty$, where $W^0$ is a Brownian bridge. This result sheds some light on the problem of describing the behavior of the eigenvectors of $M_n$ for $n$ large and for general $v_{11}$.

Article information

Source
Ann. Probab., Volume 18, Number 3 (1990), 1174-1194.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990741

Digital Object Identifier
doi:10.1214/aop/1176990741

Mathematical Reviews number (MathSciNet)
MR1062064

Zentralblatt MATH identifier
0708.62051

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 62H99: None of the above, but in this section

Keywords
Weak convergence on $D\lbrack 0, 1 \rbrack$ and $D\lbrack 0, \infty)$ eigenvectors of sample covariance matrix Brownian bridge Haar measure signed measures

Citation

Silverstein, Jack W. Weak Convergence of Random Functions Defined by The Eigenvectors of Sample Covariance Matrices. Ann. Probab. 18 (1990), no. 3, 1174--1194. doi:10.1214/aop/1176990741. https://projecteuclid.org/euclid.aop/1176990741


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