Moment inequalities for sums of weakly dependent random fields

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 δ is bounded, in Lp distance, by a known decreasing function of δ. 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.