Regularity of sub -Gaussian processes and other random fields
Abstract
This dissertation 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. We introduce a boundedness condition on the Malliavin derivative of a random variable to study sub-Gaussian and other non-Gaussian properties of functionals of random fields, with particular attention to the estimation of suprema. The boundedness of the nth Malliavin derivative is related to a new class of "sub-nth-Gaussian chaos" processes. We utilize a sharp and convenient condition using iterated Malliavin derivatives to obtain sub- nth-Gaussian chaos concentration inequalities for the expected supremum, which generalizes the Borell-Sudakov inequality. 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 as the Borell-Sudakov inequality, and for the field's modulus of continuity.
Degree
Ph.D.
Advisors
Viens, Purdue University.
Subject Area
Mathematics
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