Empirical process theory for nonsmooth functions under functional dependence

TL;DR

This paper develops an empirical process theory for nonsmooth functions applied to locally stationary processes, introducing a flexible dependence measure, and establishes convergence results for empirical distribution functions and kernel density estimators.

Contribution

It introduces a novel empirical process framework using a flexible dependence measure for nonsmooth functions in locally stationary processes.

Findings

Functional central limit theorem established

Nonasymptotic maximal inequalities derived

Uniform convergence rates for kernel density estimators obtained

Abstract

We provide an empirical process theory for locally stationary processes over nonsmooth function classes. An important novelty over other approaches is the use of the flexible functional dependence measure to quantify dependence. A functional central limit theorem and nonasymptotic maximal inequalities are provided. The theory is used to prove the functional convergence of the empirical distribution function (EDF) and to derive uniform convergence rates for kernel density estimators both for stationary and locally stationary processes. A comparison with earlier results based on other measures of dependence is carried out.