Donsker results for the empirical process indexed by functions of locally bounded variation and applications to the smoothed empirical process

Abstract

Recently, Radulovic and Wegkamp introduced a new technique to show convergence in distribution of the empirical process indexed by functions of bounded variation. This method of proof allows to directly extend convergence results known for the canonical empirical process to convergence in distribution of the empirical process indexed by functions of bounded variation. The purpose of this article is twofold. First, we extend the mentioned technique to index functions of locally bounded variation. Second, and more importantly, we demonstrate that this technique provides a new approach to show convergence in distribution of the smoothed empirical process based on kernel density estimators. Using this approach we can prove to the best of our knowledge the first results on convergence in distribution of the smoothed empirical process of dependent data. Our results cover both weak and strong dependence as well as index sets of functions of locally bounded variation. Moreover our results cover an MISE optimal choice of the bandwidth for the kernel density estimator which to some extent is the plug-in property in the Bickel–Ritov sense. In the case of i.i.d. data our results extend a seminal result of Giné and Nickl.