Understanding the time-varying structure of complex temporal systems is one of the main challenges of modern time series analysis. In this paper, we show that every uniformly-positive-definite-in-covariance and sufficiently short-range dependent non-stationary and nonlinear time series can be well approximated globally by a white-noise-driven auto-regressive (AR) process of slowly diverging order. To our best knowledge, it is the first time such a structural approximation result is established for general classes of non-stationary time series. A high dimensional $\mathcal{L}^2$ test and an associated multiplier bootstrap procedure are proposed for the inference of the AR approximation coefficients. In particular, an adaptive stability test is proposed to check whether the AR approximation coefficients are time-varying, a frequently-encountered question for practitioners and researchers of time series. As an application, globally optimal short-term forecasting theory and methodology for a wide class of locally stationary time series are established via the method of sieves.

翻译：理解复杂时间系统的时变结构是现代时间序列分析的主要挑战之一。本文证明，每个协方差一致正定且具有足够短程依赖性的非平稳非线性时间序列，均可由白噪声驱动的缓慢发散阶自回归（AR）过程全局良好逼近。据我们所知，这是首次为一般非平稳时间序列建立此类结构逼近结果。我们提出了一种高维$\mathcal{L}^2$检验及其关联乘子自助法，用于推断AR逼近系数。特别地，文中提出了自适应稳定性检验，以判断AR逼近系数是否随时间变化——这是时间序列实践者与研究者常遇到的问题。作为应用，通过筛法为广泛局部平稳时间序列建立了全局最优短期预测理论与方法。