An asymmetric Marcinkiewicz-Zygmund LLN for random fields

The classical Marcinkiewicz-Zygmund law for i.i.d. random variables has been generalized by Gut [Gut, A., 1978. Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 6, 469-482] to random fields. Therein all indices have the same power in the normalization. Looking into some weighted means of random fields, such as Cesro summation, it is of interest to generalize these laws to the case where different indices have different powers in the normalization. In this paper we give precise moment conditions for such laws.