Geodesic
Convergence of averages in the ergodic theorem for groups~$\mathbb Z^d$
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part III, Tome 256 (1999), pp. 121-128

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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. 
@article{ZNSL_1999_256_a8,
     author = {A. G. Kachurovskii},
     title = {Convergence of averages in the ergodic theorem for groups~$\mathbb Z^d$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {121--128},
     publisher = {mathdoc},
     volume = {256},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_256_a8/}
}
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A. G. Kachurovskii. Convergence of averages in the ergodic theorem for groups~$\mathbb Z^d$. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part III, Tome 256 (1999), pp. 121-128. http://geodesic.mathdoc.fr/item/ZNSL_1999_256_a8/