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

2024/12/18 by Lê Vǎn Thành, Thành, Lê Vǎn
Decision Sciences · #60F05 #60F15 #60F25 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models

paper · pdf · doi:10.48550/arxiv.2412.14016

openalex publication_date 2024/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

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--Erdös--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.

Related

• Weak and strong law of large numbers for strictly stationary Banach-valued random fields
• Strong Law of Large Numbers for Functionals of Random Fields With\n Unboundedly Increasing Covariances
• Strong laws of large numbers for arrays of random variables and stable random fields
• Almost sure convergence for the maximum of nonstationary random fields
• On the Law of Large Numbers and Convergence Rates for the Discrete Fourier Transform of Random Fields