Orthomartingale-coboundary decomposition for stationary random fields

TL;DR

This paper introduces a new projective condition for stationary random fields on multidimensional lattices, enabling their approximation by orthomartingales, which facilitates proving limit theorems and large deviations.

Contribution

It extends the martingale-coboundary decomposition to multidimensional settings, providing a powerful tool for analyzing stationary random fields.

Findings

Provides a new projective condition for approximation by orthomartingales.

Enables derivation of limit theorems for stationary random fields.

Facilitates large deviations inequalities in multidimensional contexts.

Abstract

We provide a new projective condition for a stationary real random field indexed by the lattice $\Z^d$ to be well approximated by an orthomartingale in the sense of Cairoli (1969). Ourmain result can be viewed as a multidimensional version of the martingale-coboundary decomposition method which the idea goes back to Gordin (1969). It is a powerfull tool for proving limit theorems or large deviations inequalities for stationary random fields when the corresponding result is valid for orthomartingales.