Abstract: 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 pth finite moment, with p > 2, this can range from a worst case rate of to the best case rate of .

Key words and phrases: Functional central limit theorem, functional dependence measure, Gaussian approximation, weak dependence.