Empirical process theory for locally stationary processes

TL;DR

This paper develops a new empirical process framework for locally stationary processes, extending classical results to account for time dependence and providing tools like a functional CLT and maximal inequalities.

Contribution

It introduces a novel empirical process theory for locally stationary processes using the functional dependence measure, broadening applicability beyond stationary cases.

Findings

Established a functional central limit theorem for locally stationary processes.

Developed maximal inequalities for expectations of sums in this context.

Provided uniform convergence rates for nonparametric regression with locally stationary noise.

Abstract

We provide a framework for empirical process theory of locally stationary processes using the functional dependence measure. Our results extend known results for stationary Markov chains and mixing sequences by another common possibility to measure dependence and allow for additional time dependence. Our main result is a functional central limit theorem for locally stationary processes. Moreover, maximal inequalities for expectations of sums are developed. We show the applicability of our theory in some examples, for instance we provide uniform convergence rates for nonparametric regression with locally stationary noise.