Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains

Source abstract

Locally stationary Markov chains have attracted a lot of attention in statistics in the past three decades. In this paper we prove a variety of limit theorems for partial sums generated by such chains. We first provide explicit formulas for the asymptotic mean and variance (and also higher moments), and obtain optimal convergence rates towards them. We then prove a Berry--Esseen theorem, a local central limit theorem, Edgeworth expansions, and large and moderate deviations principles. Our approach involves a parametric Perron--Frobenius theorem, which is proved using the theory of complex (Hilbert) projective metrics developed in \cite{Rugh,Dubois}, together with local approximation arguments and ideas in \cite{dolgopyat2023berry}.