  We obtain an optimal bound for a Gaussian approximation of a large class of
vector-valued random processes. Our results provide a substantial
generalization of earlier results that assume independence and/or stationarity.
Based on the decay rate of the functional dependence measure, we quantify the
error bound of the Gaussian approximation using the sample size $n$ and the
moment condition. Under the assumption of $p$th finite moment, with $p>2$, this
can range from a worst case rate of $n^{1/2}$ to the best case rate of
$n^{1/p}$.
