Doob's estimate for coherent random variables and maximal operators on trees

TL;DR

This paper extends Doob's maximal estimate to coherent random variables on trees, linking maximal functions with combinatorial properties of the Hardy-Littlewood maximal operator.

Contribution

It introduces a new version of Doob's estimate for coherent variables on trees, connecting stochastic maximal inequalities with harmonic analysis tools.

Findings

Established a Doob's maximal estimate for coherent vector variables.

Linked maximal function analysis with Hardy-Littlewood operator properties.

Provided a framework for analyzing maximal inequalities on tree structures.

Abstract

Let $\xi$ be an integrable random variable defined on $(\Omega, \mathcal{F}, \mathbb{P})$. Fix $k\in \mathbb{Z}_+$ and let $\{\mathcal{G}_{i}^{j}\}_{1\le i \le n, 1\le j \le k}$ be a reference family of sub-$\sigma$-fields of $\mathcal{F}$, such that $\{\mathcal{G}_{i}^{j}\}_{1\le i \le n}$ is a filtration for each $j\in \{1,2,\dots,k\}$. In this article we explain the underlying connection between the analysis of the maximal functions of the corresponding coherent vector and basic combinatorial properties of the uncentered Hardy-Littlewood maximal operator. Following a classical approach of Grafakos, Kinnunen and Montgomery-Smith, we establish an appropriate version of the celebrated Doob's maximal estimate.