Approximations for the maximum of a vector-valued stochastic process with drift

Giving a generalization of Berkes and Horváth (2003), we consider the Euclidean norm of vector-valued stochastic processes, which can be approximated with a vector-valued Wiener process having a linear drift. The suprema of the Euclidean norm of the processes are not far away from the norm of the processes at the right most point. We also obtain an approximation for the supremum of the weighted Euclidean norm with a Wiener process.