General theories unifying ergodic averages and martingales

The curious fact that the behavior of ergodic averages is analogous to the behavior of (reversed) martingales has long been known. For the last 60 years, at least five different approaches have been proposed to the construction of a unifying theory: by M. Jerison (1955), G.-C. Rota (1961), A. and C. Ionescu Tulcea (1963), A.M. Vershik (1960s), and A.G. Kachurovskii (1998). The aim of the present survey is to carry out a comparative analysis of all the approaches, with special focus being placed on the fifth approach, which is due to the present author and has not yet been published in full detail. Moreover, the comparative analysis carried out in the survey has led to a new, the sixth, approach.     

Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem

Mathematical Notes - Estimates of the rate of convergence in the Birkhoff ergodic theorem which hold almost everywhere are considered. For the action of an ergodic automorphism, the existence of...     

Convergence of Averages in the Ergodic Theorem for Groups ?d

Some estimates of the rates of convergence in the ergodic theorem for actions of groups ^d are given. Moreover, a martingale-ergodic theorem for ^d is proved. This theorem may be considered as an ergodic theorem in which the exact initial coordinates of points in the phase space are gradually forgotten. Bibliography: 9 titles.     

Large deviations and the rate of convergence in the Birkhoff ergodic theorem

For bounded averaged functions, we prove the equivalence of the power-law and exponential rates of convergence in the Birkhoff individual ergodic theorem with the same asymptotics of the probability of large deviations in this theorem.     

On the constants in the estimates of the rate of convergence in the Birkhoff Ergodic theorem

Mathematical Notes -