Weak Convergence of a Collection of Random Functions Defined by the Eigenvectors of Large Dimensional Random Matrices

TL;DR

This paper proves that certain functions derived from eigenvectors of large random matrices converge to Brownian bridges, extending previous results and applying to noise detection in high-dimensional data.

Contribution

It establishes weak convergence of eigenvector-based processes to Brownian bridges for a broad class of random matrices, including new matrix models.

Findings

Eigenvector functions converge to independent Brownian bridges

Results extend to real orthogonal matrices and specific matrix models

Applications include noise detection in high-dimensional sampling

Abstract

For each $n$, let $U_n$ be Haar distributed on the group of $n\times n$ unitary matrices. Let $\bfx_{n,1},\ldots,\bfx_{n,m} $ denote orthogonal nonrandom unit vectors in ${\Bbb C}^n$ and let $\text{\bf u}_{n,k}=(u_k^1,\ldots,u_k^n)^*=U^*\text{\bf x}_{n,k}$, $k=1,\ldots,m$. Define the following functions on [0,1]: $X^{k,k}_n(t)=\sqrt n\sum_{i=1}^{[nt]}(|u_k^i|^2-\tfrac1n)$, $X_n^{k,k'}(t)=\sqrt{2n}\sum_{i=1}^{[nt]}\bar u_k^iu_{k'}^i$, $k<k'$. %("$\bar{\,\,\,\,\,}$" denoting conjugate). Then it is proven that $X_n^{k,k},\Re X_n^{k,k'}$, $\Im X_n^{k,k'}$, considered as random processes in $D[0,1]$, converge weakly, as $n\to\infty$, to $m^2$ independent copies of Brownian bridge. The same result holds for the $m(m+1)/2$ processes in the real case, where $O_n$ is real orthogonal Haar distributed and $\bfx_{n,i}\in{\Bbb R}^n$, with $\sqrt n$ in $X^{k,k}_n$ and $\sqrt{2n}$ in $X_n^{k,k'}$ replaced with $\sqrt{\frac n2}$ and $\sqrt{n}$, respectively. This latter result will be shown to hold for the matrix of eigenvectors of $M_n=(1/s)V_nV_n^T$ where $V_n$ is $n\times s$ consisting of the entries of $\{v_{ij}\},\ i,j=1,2,\ldots$, i.i.d. standardized and symmetrically distributed, with each $\bfx_{n,i}=\{\pm1/\sqrt n,\ldots,\pm1/\sqrt n\}$, and $n/s\to y>0$ as $n\to\infty$. This result extends the result in J.W. Silverstein {\sl Ann. Probab. \bf18} 1174-1194. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector.