Consistency of Statistical Models in the Stationary Case.

Recently established asymptotic likelihood theory [6] concerns the study of the asymptotic behavior of various maximum estimators with the maximum likelihood estimator as the leading example. The main objective of this study is to perform the estimation of the true parameter value for the unknown distribution. The estimation is based on the observations that form a sequence of independent and identically distributed random variables. The main purpose of this paper is to investigate the case where the given observations form a stationary ergodic sequence of random variables. The first step in this direction is devoted to the foundation of the problem itself. Although dropping independence one causes some difficulties it turns out that using the results and methods established in [6] and [18] we reach our primary ambition in this direction by characterizing the sets of all accumulation and limit points of maximum estimators under consideration. Then we pass to the problem of consistency. We show that slightly stronger conditions than those established in [6] imply consistency in the present case. Moreover by using the uniform law of large numbers that is recently established in the stationary case in [19], as well as the methods developed for this purpose, we deduce new conditions implying consistency. These conditions are of eventual total boundedness in the mean type. In this way the problem of consistency of the given statistical models is naturally connected with the infinitely dimensional (uniform) law of large numbers. In particular all of the derived results apply to the maximum likelihood estimator based on the stationary ergodic observations.