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

Given an observation of the uniform empirical process α n , its functional increments α n ( u + a n ⋅ ) − α 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 n → ∞ and a n ↓ 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 the usual Poissonisation tools for the local empirical process.