Statistical properties of Markov shifts: part II-LLT

TL;DR

This paper establishes Local Central Limit Theorems for sums of functions of Markov chains under specific contraction and regularity conditions, extending previous Berry-Esseen results and addressing questions in statistical dynamics.

Contribution

It proves LLT for Markov chains with equicontinuous transition probabilities and bounded densities, complementing earlier Berry-Esseen results and answering open questions.

Findings

Proved LLT for Markov chains with equicontinuous transition probabilities.

Extended previous Berry-Esseen theorems to new classes of functions.

Addressed open questions in statistical dynamics regarding local limit behaviors.

Abstract

We prove Local Central Limit Theorems (LLT) for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j(...,X_{j-1},X_j,X_{j+1},...)$, where $(X_j)$ is a Markov chains with equicontinuous conditional probabilities satisfying contraction conditions close in spirit to Dobrushin's, and some ``physicality" assumptions and $f_j$ are equicontinuous functions. Our conditions will always be in force when the chain takes values on a metric space and have uniformly bounded away from $0$ backward transition densities with respect to a measure which assigns uniform positive mass to certain ``balls". This paper complements \cite{MarShif1} where Berry-Esseen theorems, were proven for (not necessarily continuous) functions satisfying certain approximation conditions. Our results address a question posed by D. Dolgopyat and O. Sarig in \cite[Section 1.5]{DS}.