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

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.