Statistical properties of Markov shifts: part II-LLT
Abstract
We prove Local Central Limit Theorems (LLT) for partial sums of the form Sn=Σj=0n-1fj(...,Xj-1,Xj,Xj+1,...), where (Xj) is a Markov chains with equicontinuous conditional probabilities satisfying contraction conditions close in spirit to Dobrushin's, and some ``physicality" assumptions and fj 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 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 [Section 1.5]DS.
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