On the functional CLT for stationary Markov Chains started at a point

TL;DR

This paper establishes a general functional central limit theorem for stationary Markov chains started at a point, introducing new classes of processes satisfying this result and extending to strongly mixing sequences.

Contribution

It provides a new quenched functional CLT applicable to stationary Markov chains and related processes, with simplified proofs for strongly mixing sequences.

Findings

The CLT applies to reversible, ergodic Markov chains with bounded additive functionals.

New classes of stationary processes satisfying the quenched CLT are identified.

The approach offers a shorter proof for certain results in strongly mixing sequences.

Abstract

We present a general functional central limit theorem started at a point also known under the name of quenched. As a consequence, we point out several new classes of stationary processes, defined via projection conditions, which satisfy this type of asymptotic result. One of the theorems shows that if a Markov chain is stationary ergodic and reversible, this result holds for bounded additive functionals of the chain which have a martingale coboundary in L_1 representation. Our results are also well adapted for strongly mixing sequences providing for this case an alternative, shorter approach to some recent results in the literature.