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arXiv · 2510.24244

Statistical properties of Markov shifts: part II-LLT

Yeor Hafouta
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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}.

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Yeor Hafouta. 2025-10-28. Statistical properties of Markov shifts: part II-LLT. https://arxiv.org/abs/2510.24244

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