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

TL;DR

This paper investigates weak dependence properties of local functionals of Markov chains on bZ^d, establishing conditions under which a Central Limit Theorem applies, especially for functions related to Hflder continuous functions on the torus.

Contribution

It introduces a class of local functions for Markov chains in bZ^d, demonstrating CLT applicability based on spectral gap and Hflder regularity.

Findings

Existence of spectral gap for certain local functions.

CLT holds for functions with sufficient Hflder regularity.

Weak dependence results extend to Markov chains in bZ^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 $f(\widehat \eta)$, where $\widehat \eta= \{\eta_{t}\}_{t=0}^{\infty}$ 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 H\"older 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 $f(S^{k}\widehat \eta)$, $k=0,1,\ldots$, where $S$ is the time shift.