Strong laws of large numbers for arrays of random variables and stable random fields

TL;DR

This paper establishes strong laws of large numbers for dependent random fields, including those with heavy tails like fractional stable fields, using moment-based conditions and a novel maximal inequality.

Contribution

It extends SLLN to random fields with dependence and heavy tails, introducing a new maximal inequality for partial sums of such fields.

Findings

SLLN holds under moment conditions for dependent random fields

Applicable to heavy-tailed distributions including fractional stable fields

Provides a new maximal inequality for moments of partial sums

Abstract

Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the $p$-th moments of the partial sums of the random fields, which are convenient to verify. The main technical tool in this paper is a maximal inequality for the moments of partial sums of random fields that extends the technique of Levental, Chobanyan and Salehi \cite{chobanyan-l-s} for a sequence of random variables indexed by a one-parameter.