A central limit theorem for fields of martingale differences

TL;DR

This paper establishes a central limit theorem for fields of martingale differences generated by commuting transformations, extending known results from Bernoulli fields to more general ergodic systems.

Contribution

It generalizes the CLT for martingale difference fields from Bernoulli to ergodic systems with commuting transformations, requiring only ergodicity of one transformation.

Findings

Proves a CLT for fields generated by commuting transformations.

Extends CLT results from Bernoulli to ergodic systems.

Requires only ergodicity of one transformation.

Abstract

We prove a central limit theorem for a random field generated by d commuting probability preserving transformations; the martingale is given by a commuting filtration (cf. D. Khosnevisan, Multiparameter Processes, Springer 2002). The result has been known for Bernoulli random fields. Here, only ergodicity of one of generating transformations is supposed.