Gaussian approximation for nonstationary time series with optimal rate and explicit construction

Abstract

Statistical inference for time series such as curve estimation for time-varying models or testing for existence of a 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 nonstationary 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 nonstationary 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 nonstationary 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 nonstationary errors. Finally, we substantiate our theoretical results with simulation studies and real data analysis.