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We derive both Azuma-Hoeffding and Burkholder-type inequalities for partialsums over a rectangular grid of dimension $d$ of a random field satisfying aweak dependency assumption of projective type: the difference between theexpectation of an element of the random field and its conditional expectationgiven 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 isbased on the combination of a multi-scale approximation of random sums bymartingale difference sequences, and of a careful decomposition of the domain.The obtained results extend previously known bounds under comparablehypotheses, and do not use the assumption of commuting filtrations.

Moment inequalities for sums of weakly dependent random fields