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

TL;DR

This paper studies the almost sure limit behavior of multivariate functional increments of the uniform empirical process, establishing convergence and limit laws using extended Poissonisation techniques.

Contribution

It introduces new almost sure results for the functional increments of the uniform empirical process and extends Poissonisation methods for local empirical processes.

Findings

Convergence in distribution of the functional increments.

Functional limit laws are established.

Results hold under mild conditions on the increment scale.

Abstract

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.