Convergence theorems and maximal inequalities for martingale ergodic processes

In this article, we study two types of martingale ergodic processes. We prove that a.e. convergence and Lp convergence as well as maximal inequalities, which are established both in ergodic theory and martingale setting, also hold well for these new sequences of random variables. Moreover, the corresponding theorems in the former two areas turn out to be degenerate cases of the martingale ergodic theorems proved here.