Weak dependence for a class of local functionals of Markov chains on ${\mathbb{Z}}^d$

Abstract

In some papers on infinite Markov chains in ${\mathbb{Z}}^d$, and notably in the work of R.A. Minlos and collaborators, one can prove the existence of a spectral gap for a suitable subspace of local functions. We consider functions of the type $f(\hat{\eta })$, where $\hat{\eta }=\{{\eta }_t{\}}_{t=0}^{\infty }$ is the sequence of the states, and $f$ is local. In the case of a simple example of random walk in random environment with mutual interaction we show that there is a natural class of functions $f$, related to the Hölder continuos functions ${\mathcal{C}}^{\alpha }$on the torus $T^1$, with $\alpha \in (0,1)$ large enough, depending on the spectral gap, for which the Central Limit Theorem holds for the sequence $f(S^k\hat{\eta })$, $k=0,1,\dots$, where $S$ is the time shift.

15 pages; important change: new version of the proof of Theorem 4.2