  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.
