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]: $X^{k,k}_n(t)=\sqrt n\sum_{i=1}^{[nt]}(|u_k^i|^2-\tfrac1n)$, $X_n^{k,k'}(t)=\sqrt{2n}\sum_{i=1}^{[nt]}\bar u_k^iu_{k'}^i$, $k 0$ 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.