  We use Stein's method to obtain a bound on the distance between scaled
$p$-dimensional random walks and a $p$-dimensional (correlated) Brownian
Motion. We consider dependence schemes including those in which the summands in
scaled sums are weakly dependent and their $p$ components are strongly
correlated. As an example application, we prove a functional limit theorem for
exceedances in an $m$-scans process, together with a bound on the rate of
convergence. We also find a bound on the rate of convergence of scaled
U-statistics to Brownian Motion, representing an example of a sum of strongly
dependent terms.
