Uniform central limit theorems for non-stationary processes via relative weak convergence
Nicolai Palm, Thomas Nagler
TL;DR
The paper tackles the breakdown of classical CLTs for non-stationary data by introducing relative weak convergence, which compares evolving statistics to a sequence of Gaussian processes. It develops relative central limit theorems for finite- and infinite-dimensional statistics, including weighted and sequential empirical processes, under moment, mixing, and bracketing-entropy conditions. Bootstrap inference is adapted to this setting via multiplier and block schemes, with proven consistency relative to the evolving Gaussian limits. The framework enables plug-in, nonparametric inference (e.g., uniform confidence bands) and hypothesis testing for non-stationary time series, with concrete applications to nonparametric trend estimation and time-series characteristics 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.