  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.
