Statistical properties of Markov shifts: part II-LLT

We prove Local Central Limit Theorems (LLT) for partial sums of the form $S_n={\sum }_{j=0}^{n-1}{f}_j(\mathrm{.}\mathrm{.}\mathrm{.},X_{j-1},X_j,X_{j+1},\mathrm{.}\mathrm{.}\mathrm{.})$, 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}.