The now classical convergence in distribution theorem for well normalized sums of stationary martingale increments has been extended to multi-indexed martingale increments (see Voln\'{y} (2019) and references in there). In the present article 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 still the 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 associated to the stationary random field allows us to give a necessary and sufficient condition for 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