Global Central Limit Theorems for Markov chains with normal transition operator

Christophe Cuny; Michael Lin

paper

Global Central Limit Theorems for Markov chains with normal transition operator

arXiv:2610.02282

Abstract

Let $P$ be a Markov operator on a general state space $(S, Σ)$ with invariant probability $m$ , assumed ergodic. When $P$ is normal in $L_{2} (m)$ , we prove the necessity of uniform ergodicity, under some spectral conditions (satisfied when $P$ is symmetric), for the annealed CLT to hold for {\it every} centered $f \in L_{2} (m)$ . We also obtain results without assuming normality under some Ritt-type conditions.