Some new almost sure results on the functional increments of the uniform empirical process

Given an observation of the uniform empirical process αn, its functional increments αn(u+an⋅)−αn(u) can be viewed as a single random process, when u is distributed under the Lebesgue measure. We investigate the almost sure limit behaviour of the multivariate versions of these processes as n→∞ and an↓0. Under mild conditions on an, a convergence in distribution and functional limit laws are established. The proofs rely on a new extension of the usual Poissonisation tools for the local empirical process.