Ces\'aro summation for random fields

TL;DR

This paper extends Cesàro summation methods and convergence results from sequences of random variables to two-dimensional random fields indexed by positive integer lattice points.

Contribution

It generalizes Cesàro summation and convergence theorems from sequences to random fields on -dimensional lattices, including almost sure and weak convergence.

Findings

Extended Cesàro summation results to random fields.

Proved almost sure convergence of Cesàro means for D random fields.

Established weak laws and complete convergence for these fields.

Abstract

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.