Asymptotic Expansions for Stochastic Processes

The central limit theorems are the basis of the large sample statistics. In estimation theory, the asymptotic efficiency is evaluated by the asymptotic variance of estimators, and in testing statistical hypotheses, the critical region of a test is determined by the normal approximation. Though asymptotic properties of statistics are based on central limit theorems, the accuracy of their approximation is not necessarily sufficient in practice, especially in the case not many observations are available. Even then, we experienced possibility of getting more precise approximation by the asymptotic expansion methods. The asymptotic expansion has theoretical importance. This method is today recognized as basis of various branches of theoretical statistics like higher order inferential theory, prediction, model selection, resampling methods, information geometry, and so on. For example, the Akaike Information Criterion (AIC) for statistical model selection is a statistic that incorporates higher-order behavior of the maximum log likelihood. In the recent four decades, intensive studies have been done for statistics of semimartingales. See, e.g., Kutoyants [54, 55, 56], Basawa and Prakasa Rao [8], Küchler and Soerensen [51], and Prakasa Rao [80, 79]. Since large sample theoretical approaches are inevitable to semimartingales, the development was in exact timing interactively with that of limit theorems. The counterpart of traditional independent observations is the class of stochastic processes with ergodic property. Laws of large numbers were often deduced from mixing properties or from ergodic theorems through Markovian structures of processes, and various central limit theorems have been produced in the mixing framework and in the martingale framework. Thus, after developments of the first order statistics, it was natural that