Martingale ergodic theorem
type: Item
instance of: scholarly article
title: Martingale ergodic theorem
zbMATH Open document ID: 0922.60045
DOI: 10.1007/BF02310315
author: A. G. Kachurovskii
published in: Mathematical Notes
publication date: 1999-10-14T00:00:00Z
review text: An interesting coincidence in the behaviour of ergodic means and (reversed) martingales has long been known. The author introduces a natural class of stochastic processes called the martingale ergodic processes. The sequences of random variables belonging to this class converge both almost everywhere and in norm (as it is proved in Theorem 1). Then the classical Birkhoff and von Neumann ergodic theorems, as well as the Doob theorems on convergence of direct and reversed martingales appear as particular (degenerated) cases of Theorem 1. Furthermore, the maximal and dominant inequalities are valid for the martingale ergodic sequences as well. From a physical viewpoint, the martingale ergodic theorem can be regarded as an ergodic theorem of the phase space gradually forgetting their exact initial coordinates.
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 1274245
zbMATH Keywords: martingale ergodic processes
zbMATH Keywords: direct and reversed martingales
zbMATH Keywords: phase space
reviewed by: Sophia L. Kalpazidou
MaRDI profile type: Publication
cites work: An upcrossing inequality with applications
cites work: Q4114574
cites work: Q3994694
description: scientific article; zbMATH DE number 1274245