Orlicz integrability of additive functionals of Harris ergodic Markov chains

TL;DR

This paper establishes optimal bounds for Orlicz norms of additive functionals of Harris ergodic Markov chains, linking tail behavior and limit theorems to the chain's regeneration structure.

Contribution

It provides the first optimal Orlicz norm estimates for sums of functions along Harris ergodic Markov chains, connecting tail estimates and classical limit theorems.

Findings

Optimal Orlicz norm bounds for additive functionals

Applications to tail estimates for unbounded functions

Extensions to CLT, LIL, and Berry-Esseen theorems

Abstract

For a Harris ergodic Markov chain $(X_n)_{n\ge 0}$, on a general state space, started from the so called small measure or from the stationary distribution we provide optimal estimates for Orlicz norms of sums $\sum_{i=0}^\tau f(X_i)$, where $\tau$ is the first regeneration time of the chain. The estimates are expressed in terms of other Orlicz norms of the function $f$ (wrt the stationary distribution) and the regeneration time $\tau$ (wrt the small measure). We provide applications to tail estimates for additive functionals of the chain $(X_n)$ generated by unbounded functions as well as to classical limit theorems (CLT, LIL, Berry-Esseen).