Comment on a theorem of M. Maxwell and M. Woodroofe

TL;DR

This paper provides a direct derivation of Maxwell and Woodroofe's theorem on martingale approximation for additive functionals of stationary Markov processes, based on the non-reversible Kipnis-Varadhan theorem.

Contribution

It offers a new, straightforward derivation of a key theorem in Markov process theory, connecting it to the non-reversible Kipnis-Varadhan result.

Findings

Derived a direct proof of Maxwell and Woodroofe's theorem

Linked martingale approximation to non-reversible Kipnis-Varadhan theorem

Clarified the theoretical foundation for additive functionals in Markov processes

Abstract

We present a direct derivation of the theorem of M. Maxwell and M. Woodroofe (Ann. Probab. 28 (2000) 713-724), on martingale approximation of additive functionals of stationary Markov processes, from the non-reversible version of the Kipnis-Varadhan theorem.