Fuk's inequalities for stochastic fields of reversed martingales (Q1177564)

Let \(\{X_ n, n\in\mathbb{Z}^ d\}\) be a family of random variables adapted to a backward filtration \(\{{\mathcal F}_ n, n\in \mathbb{Z}^ d\}\) whereas the usual partial ordering in \(\mathbb{Z}^ d\) is adopted. Let, for any \(m,n\in\mathbb{Z}^ d\) with \(m<n\), \(E(X_ m\mid {\mathcal F}_ n)=0\) a.s. hold. Several inequalities for the backward martingale \(S_ n=\sum_{k\geq n}X_ k\) are proved.