Uniform central limit theorems for non-stationary processes via relative weak convergence

TL;DR

This paper introduces relative weak convergence, a new framework extending classical CLTs to non-stationary processes, enabling statistical inference where traditional methods fail due to non-stationarity.

Contribution

The paper develops relative weak convergence and establishes new relative CLTs applicable to non-stationary data, broadening the scope of asymptotic theory.

Findings

Established concrete relative CLTs for random vectors and empirical processes.

Extended CLT variants include sequential, weighted, and bootstrap methods.

Demonstrated applications in nonparametric trend estimation and hypothesis testing.

Abstract

Statistical inference for non-stationary data is hindered by the failure of classical central limit theorems (CLTs), not least because there is no fixed Gaussian limit to converge to. To resolve this, we introduce relative weak convergence, an extension of weak convergence that compares a statistic or process to a sequence of <evolving processes. Relative weak convergence retains the essential consequences of classical weak convergence and coincides with it under stationarity. Crucially, it applies in general non-stationary settings where classical weak convergence fails. We establish concrete relative CLTs for random vectors and empirical processes, along with sequential, weighted, and bootstrap variants that parallel the state-of-the-art in stationary settings. Our framework and results offer simple, plug-in replacements for classical CLTs whenever stationarity is untenable, as illustrated by applications in nonparametric trend estimation and hypothesis testing.