On Rio's proof of limit theorems for dependent random fields

TL;DR

This paper extends Rio's proof technique to dependent random fields, establishing several classical limit theorems with improved conditions and new maximal inequalities for broad dependence structures.

Contribution

It introduces a unified approach to prove limit theorems for dependent random fields, extending Rio's method and deriving new maximal inequalities.

Findings

Proved the Hsu–Robbins–Erdős–Spitzer–Baum–Katz theorem for dependent fields.

Established the Feller weak law of large numbers for dependent random fields.

Derived the Pyke–Root theorem on mean convergence under general dependence.

Abstract

This paper presents an exposition of Rio's proof of the strong law of large numbers and extends his method to random fields. In addition to considering the rate of convergence in the Marcinkiewicz--Zygmund strong law of large numbers, we go a step further by establishing (i) the Hsu--Robbins--Erd\"{o}s--Spitzer--Baum--Katz theorem, (ii) the Feller weak law of large numbers, and (iii) the Pyke--Root theorem on mean convergence for dependent random fields. These results significantly improve several particular cases in the literature. The proof is based on new maximal inequalities that hold for random fields satisfying a very general dependence structure.