  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
