Explore a comprehensive lecture on auto-regressive approximations to non-stationary time series, focusing on inference and applications. Delve into the challenges of modern time series analysis, particularly in understanding time-varying structures of complex temporal systems. Discover how short-range dependent non-stationary and nonlinear time series can be globally approximated using white-noise-driven auto-regressive (AR) processes with slowly diverging orders. Examine uniform statistical inference of AR structures through high-dimensional L2 tests. Investigate practical applications of AR approximation theory, including globally optimal short-term forecasting, efficient estimation, and resampling inference under complex temporal dynamics. Gain valuable insights into advanced time series analysis techniques and their real-world implications in this 57-minute presentation by Zhou Zhou at the Colloque des sciences mathématiques du Québec (CSMQ).