Orlicz Integrability of Additive Functionals of Harris Ergodic Markov Chains

Abstract

For a Harris ergodic Markov chain (Xn)n ≥ 0, on a general state space, started from the small measure or from the stationary distribution, we provide optimal estimates for Orlicz norms of sums ∑i = 0τf(Xi), where τ is the first regeneration time of the chain. The estimates are expressed in terms of other Orlicz norms of the function f (with respect to the stationary distribution) and the regeneration time τ (with respect to the small measure). We provide applications to tail estimates for additive functionals of the chain (Xn) generated by unbounded functions as well as to classical limit theorems (CLT, LIL, Berry-Esseen).