Moment inequalities for sums of weakly dependent random fields

TL;DR

This paper develops new moment inequalities for sums of weakly dependent random fields on a grid, extending existing bounds without requiring commuting filtrations, by combining multi-scale approximation and domain decomposition.

Contribution

It introduces Azuma-Hoeffding and Burkholder-type inequalities for weakly dependent random fields, generalizing prior results and removing the need for commuting filtrations.

Findings

Derived new inequalities for weakly dependent random fields.

Extended bounds to higher dimensions and weaker dependency assumptions.

Eliminated the requirement of commuting filtrations in the analysis.

Abstract

We derive both Azuma-Hoeffding and Burkholder-type inequalities for partial sums over a rectangular grid of dimension $d$ of a random field satisfying a weak dependency assumption of projective type: the difference between the expectation of an element of the random field and its conditional expectation given the rest of the field at a distance more than $\delta$ is bounded, in $L^p$ distance, by a known decreasing function of $\delta$. The analysis is based on the combination of a multi-scale approximation of random sums by martingale difference sequences, and of a careful decomposition of the domain. The obtained results extend previously known bounds under comparable hypotheses, and do not use the assumption of commuting filtrations.