Functional CLT for martingale-like nonstationary dependent structures

TL;DR

This paper develops non-stationary martingale methods to establish a functional central limit theorem for various dependent, non-stationary processes, broadening the scope of classical CLT results.

Contribution

It introduces a non-stationary projective Maxwell-Woodroofe condition, enabling maximal inequalities and CLTs for complex dependent structures.

Findings

Maximal inequalities for non-stationary dependent data

Functional CLT for non-stationary -mixing sequences

CLT results for functions of non-stationary linear processes

Abstract

In this paper we develop non-stationary martingale techniques for dependent data. We shall stress the non-stationary version of the projective Maxwell-Woodroofe condition, which will be essential for obtaining maximal inequalities and functional central limit theorem for the following examples: nonstationary \r{ho}-mixing sequences, functions of linear processes with non-stationary innovations, quenched version of the functional central limit theorem for a stationary sequence, evolutions in random media such as a process sampled by a shifted Markov chain.