Supremum Concentration Inequality and Modulus of Continuity for Sub-nth Chaos Processes

This article provides a detailed analysis of the behavior of suprema and moduli of continuity for a large class of random fields which generalize Gaussian processes, sub-Gaussian processes, and random fields that are in the nth chaos of a Wiener process. An upper bound of Dudley type on the tail of the random field’s supremum is derived using a generic chaining argument; it implies similar results for the expected supremum, and for the field’s modulus of continuity. We also utilize a sharp and convenient condition using iterated Malliavin derivatives, to arrive at similar conclusions for suprema, via a different proof, which does not require full knowledge of the covariance structure.