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Central limit theorems for additive functionals of reversible Markov chains and applications. (English) Zbl 0584.60043

Colloq. Honneur L. Schwartz, Éc. Polytech. 1983, Vol. 2, Astérisque 132, 65-70 (1985).
[For the entire collection see Zbl 0566.00010.]
The authors consider translation invariant ergodic distributions on the set of subsets of \({\mathbb{Z}}^ k\). These random sets are considered as a set of scatterers among the sites of \({\mathbb{Z}}^ k\). They consider the trajectory process of a particle starting in zero moving rectangular on \({\mathbb{Z}}^ k\) and changing direction as in symmetrical random walks whenever it hits a scatterer.
They prove a CLT for additive functionals of reversible Markov processes via a Martingale-difference approximation argument and apply this result to prove that the position of the particle after n steps normalized by a factor 1/\(\sqrt{n}\) is approximately Gaussian.
Reviewer: F.Götze

MSC:

60F17 Functional limit theorems; invariance principles
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J55 Local time and additive functionals

Citations:

Zbl 0566.00010