Lan Based Asymptotic Theory for Time Series

The local asymptotic normality (LAN) , introduced by LeCam, is the most fundamental concept in the statistical asymptotic theory. If LAN property for a class of statistical models is established, then the asymptotic optimality of estimator and test can be described in terms of the central sequence. This concept gives a unified view for the statistical estimation and testing theory. Recently the LAN concept has been introduced to the asymptotic theory for time series. This paper provides a personal overview of the LAN results for linear processes, nonlinear processes, diffusion processes, long-memory processes, and locally stationary processes, etc.. The results are applied to the asymptotic estimation, testing theory, and discriminant analysis in time series. Then, construction of asymptotically optimal estimator, test and discriminator is discussed.