Convergence of averages in the ergodic theorem for groups $\mathbb Z^d$

Some estimates of the rates of convergence in the ergodic theorem for actions of groups $\mathbb Z^d$ are given. Besides, martingale–ergodic theorem for $\mathbb Z^d$ is proved. This theorem may be considered as an ergodic theorem in which the exact initial coordinates of phase space's points are gradually forgotten.