Ces\'aro summation for random fields

Various methods of summation for divergent series of real numbers have been generalized to analogous results for sums of iid random variables. The natural extension of results corresponding to Ces\`aro summation amounts to proving almost sure convergence of the Ces\`aro means. In the present paper we extend such results as well as weak laws and results on complete convergence to random fields, more specifically to random variables indexed by $\mathbb{Z}_+^2$, the positive two-dimensional integer lattice points.