Martingale-Coboundary Representation for a Class of Random Fields

TL;DR

This paper extends the martingale-coboundary decomposition to multivariate random fields generated by commuting transformations, facilitating limit theorem proofs for such complex stochastic processes.

Contribution

It introduces a multivariate martingale-coboundary representation for random fields, generalizing previous univariate results and handling mixed-type summands with directional properties.

Findings

Decomposition involves reversed multiparameter martingale differences.

Representation applies to random fields from commuting transformations.

Framework supports future limit theorem applications.

Abstract

A stationary random sequence admits under some assumptions a representation as the sum of two others: one of them is a martingale difference sequence, and another is a so-called coboundary. Such a representation can be used for proving some limit theorems by means of the martingale approximation. A multivariate version of such a decomposition is presented in the paper for a class of random fields generated by several commuting non-invertible probability preserving transformations. In this representation summands of mixed type appear which behave with respect to some groupof directions of the parameter space as reversed multiparameter martingale differences (in the sense of one of several known definitions) while they look as coboundaries relative to the other directions. Applications to limit theorems will be published elsewhere.