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

Given an observation of the uniform empirical process $\alp_n$, its functional increments $\alp_n(u+a_n\cdot)-\alp_n(u)$ 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 $\nif$ and $a_n\downarrow 0$. Under mild conditions on $a_n$, a convergence in distribution and functional limit laws are established. The proofs rely on a new extension of usual Poissonisation tools for the local empirical process.