Gaussian Approximation For Non-stationary Time Series with Optimal Rate and Explicit Construction

TL;DR

This paper develops explicit Gaussian approximation methods for non-stationary time series, achieving optimal rates and enabling practical statistical inference like change-point detection.

Contribution

It introduces two constructive approaches for Gaussian approximation in non-stationary series, improving applicability and theoretical understanding.

Findings

Achieves optimal convergence rates for Gaussian approximation.

Provides practical methods for non-stationary time series analysis.

Demonstrates effectiveness through simulations and real data.

Abstract

Statistical inference for time series such as curve estimation for time-varying models or testing for existence of change-point have garnered significant attention. However, these works are generally restricted to the assumption of independence and/or stationarity at its best. The main obstacle is that the existing Gaussian approximation results for non-stationary processes only provide an existential proof and thus they are difficult to apply. In this paper, we provide two clear paths to construct such a Gaussian approximation for non-stationary series. While the first one is theoretically more natural, the second one is practically implementable. Our Gaussian approximation results are applicable for a very large class of non-stationary time series, obtain optimal rates and yet have good applicability. Building on such approximations, we also show theoretical results for change-point detection and simultaneous inference in presence of non-stationary errors. Finally we substantiate our theoretical results with simulation studies and real data analysis.