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 
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/
@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},
     year = {1999},
     volume = {256},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_256_a8/}
}
TY  - JOUR
AU  - A. G. Kachurovskii
TI  - Convergence of averages in the ergodic theorem for groups $\mathbb Z^d$
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 1999
SP  - 121
EP  - 128
VL  - 256
UR  - http://geodesic.mathdoc.fr/item/ZNSL_1999_256_a8/
LA  - ru
ID  - ZNSL_1999_256_a8
ER  - 
%0 Journal Article
%A A. G. Kachurovskii
%T Convergence of averages in the ergodic theorem for groups $\mathbb Z^d$
%J Zapiski Nauchnykh Seminarov POMI
%D 1999
%P 121-128
%V 256
%U http://geodesic.mathdoc.fr/item/ZNSL_1999_256_a8/
%G ru
%F ZNSL_1999_256_a8

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] U. Krengel, Ergodic theorems, Walter de Gruyter, Berlin–New York, 1985 | MR | Zbl

[2] A. G. Kachurovskii, “Skorosti skhodimosti v ergodicheskikh teoremakh”, Usp. Mat. Nauk, 51 (1996), 73–124 | MR | Zbl

[3] A. M. Vershik, A. G. Kachurovskii, Skorosti skhodimosti v ergodicheskikh teoremakh dlya lokalno konechnykh grupp i obraschennye martingaly, Preprint POMI 2/1998. 6 s. | Zbl

[4] A. G. Kachurovskii, “Martingalno-ergodicheskaya teorema”, Mat. Zametki, 64 (1998), 311–314 | MR | Zbl

[5] I. A. Ibragimov, Yu. V. Linnik, Nezavisimye i statsionarno svyazannye velichiny, Nauka, M., 1965

[6] M. I. Dyachenko, “Nekotorye problemy teorii kratnykh trigonometricheskikh ryadov”, Usp. Mat. Nauk, 47 (1992), 97–162 | MR | Zbl

[7] A. N. Shiryaev, Veroyatnost, Nauka, M., 1989 | MR

[8] J. Neveu, Discrete parameter martingales, North Holland Publishing Co., Amsterdam, 1975 | MR | Zbl

[9] M. Loev, Teoriya veroyatnostei, IL, M., 1962 | MR