Relaxed sector condition

TL;DR

This paper introduces the relaxed sector condition (RSC), a new sufficient criterion for martingale approximation and CLT in Markov processes, generalizing existing sector conditions with simpler proofs and potential broader applications.

Contribution

The paper proposes the relaxed sector condition (RSC), extending previous sector conditions, with more transparent proofs and potential for wider applicability in Markov process analysis.

Findings

RSC generalizes SSC and GSC for Markov processes.

Simplifies proof structure compared to previous sector conditions.

Potential for applications beyond existing sector conditions.

Abstract

In this note we present a new sufficient condition which guarantees martingale approximation and central limit theorem a la Kipnis-Varadhan to hold for additive functionals of Markov processes. This condition which we call the relaxed sector condition (RSC) generalizes the strong sector condition (SSC) and the graded sector condition (GSC) in the case when the self-adjoint part of the infinitesimal generator acts diagonally in the grading. The main advantage being that the proof of the GSC in this case is more transparent and less computational than in the original versions. We also hope that the RSC may have direct applications where the earlier sector conditions don't apply. So far we don't have convincing examples in this direction.