  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.
