A Note on the Approximation of the Uniform Empirical Process

Abstract

Mason and van Zwet gave an approximation of the uniform empirical process by a Brownian bridge. Their result is a refinement of a result of Komlos, Major and Tusnady in the case when the process is considered in a small interval. In this note we show that in such cases a much better Poissonian approximation is possible which seems to be better applicable in certain cases. We also prove a multidimensional version of this result, where a sequence of uniform empirical processes is simultaneously approximated by partial sums of independent Poisson processes.