A new CLT for additive functionals of Markov chains

TL;DR

This paper establishes a new central limit theorem for additive functionals of stationary Markov chains, using a novel conditioning approach, applicable even without irreducibility or aperiodicity assumptions.

Contribution

Introduces a new method involving conditioning on both past and future to prove CLT for Markov chains with general state spaces, relaxing traditional assumptions.

Findings

CLT holds for additive functionals with bounded variance growth.

Random centering is unnecessary under stronger ergodicity conditions.

Convergence may occur to non-normal distributions without ergodicity.

Abstract

In this paper we study the central limit theorem for additive functionals of stationary Markov chains with general state space by using a new idea involving conditioning with respect to both the past and future of the chain. Practically, we show that any additive functionals of a stationary and totally ergodic Markov chain with the variance of S_n divided by n uniformly bounded, satisfies a square root central limit theorem with a random centering. We do not assume that the Markov chain is irreducible and aperiodic. However, the random centering is not needed if the Markov chain satisfies stronger forms of ergodicity. In absence an ergodicity the convergence in distribution still holds, but the limiting distribution might not be normal.