Truncations of Haar distributed matrices, traces and bivariate Brownian bridges

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