  Let U be a Haar distributed unitary matrix in U(n)or O(n). We show that after
centering the double index process $$ W^{(n)} (s,t) = \sum_{i \leq \lfloor ns
\rfloor, j \leq \lfloor nt\rfloor} |U_{ij}|^2 $$ converges in distribution to
the bivariate tied-down Brownian bridge. The proof relies on the notion of
second order freeness.
