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On conditions for applicability of the central limit theorem to random walks on a lattice. (English. Russian original) Zbl 0603.60020
Sov. Math., Dokl. 30, 410-413 (1984); translation from Dokl. Akad. Nauk SSSR 278, 531-534 (1984).
The goal of this paper is to provide sufficient conditions for applicability of the central limit theorem to Markov chains with phase space $${\mathbb{Z}}^ d$$, where the transition probabilities are realizations of a homogeneous random field. Results, which generalize those obtained by G. F. Lawler [Commun. Math. Phys. 87, 81-87 (1982; Zbl 0502.60056)] cover one strong law of large numbers and one (weak) limiting approximation of the paths by a Wiener process.
The described Markov chains are thought of as being Markov random walks in some random environment. In this paper it is assumed that the environment is statistically homogeneous, so that it can be described by a measure preserving dynamical system (transformation).
Reviewer: F.T.Bruss

MSC:
 60F05 Central limit and other weak theorems 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60G50 Sums of independent random variables; random walks 60G60 Random fields