Martingale ergodic theorem
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.
scientific article; zbMATH DE number 1274245