Moment convergence of $Z$-estimators and $Z$-process method for change point problems

Abstract: The problem to establish not only the asymptotic distribution results for statistical estimators but also the moment convergence of the estimators has been recognized as an important issue in advanced theories of statistics. One of the main goals of this paper is to present a metod to derive the moment convergence of $Z$-estimators as it has been done for $M$-estimators. Another goal of this paper is to develop a general, unified approach, based on some partial estimation functions which we call "$Z$-process", to the change point problems for ergodic models as well as some models where the Fisher information matrix is random and inhomogeneous in time. Applications to some diffusion process models and Cox's regression model are also discussed.