  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.
