Optimal Poisson approximation of uniform empirical processes

In this paper, we discuss the optimality of Poisson approximation of uniform empirical processes of size n in a small interval [0, l], in the sense that the sup-norm distance between their paths has minimum expectation. Two optimal constructions are considered. The first one depends on [0, l] and makes sense if and only if l = o(n−12), whereas the second one does not, and makes sense if and only if l = o(n−1). In both cases, we obtain the exact probability that the paths of the two processes coincide on [0, l] as well as, under appropriate assumptions, the exact order of convergence of the tail probabilities concerning the sup-norm distance between their paths. We use elementary coupling techniques which allow us to give short and simple proofs.