What can be the limit in the CLT for a field of martingale differences?

TL;DR

This paper investigates the limits of the Central Limit Theorem for multidimensional martingale differences, identifying conditions under which the limit law is normal or non-normal based on ergodic properties and entropy.

Contribution

It advances understanding of the CLT for multidimensional martingales by characterizing when the limit law is normal or non-normal using ergodic theory and entropy conditions.

Findings

Normal limit law when the process is ergodic

Non-normal limits can occur in the general case

Entropy characterizes the existence of non-normal limits

Abstract

The now classical convergence in distribution theorem for well normalized sums ofstationary martingale increments has been extended to multi-indexed martingaleincrements (see Voln\'{y} (2019) and references in there). In the presentarticle we make progress in the identification of the limit law.In dimension one, as soon as the stationary martingale increments form an ergodic process, the limit law is normal, and it is stillthe case for multi-indexed martingale increments when one of the processes defined by one coordinate of the{\it multidimensional time} is ergodic. In the general case, the limit may be non normal.The dynamical properties of the $\mathbb{Z}^d$-measure preserving action associatedto the stationary random field allows us to give a necessary and sufficient conditionfor the existence of a non-normal limit law, in terms of entropy of some random processes.The identification of a {\it natural} factor on which the $\mathbb{Z}^d$-action is {\it of product type