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

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