Problems in high-dimensional probability
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In Chapter 1, we focus on lower estimates for suprema of stochastic processes. We investigate the role of super-gaussianity and the Sudakov Minoration Principle for this concept. We derive an extension of celebrated Majorizing Measures Theorem (MMT) for Gaussian mixtures which in turn yields a comparison inequality between the expected suprema of sub-gaussian random vectors and Gaussian mixtures. In Chapter 2, we focus on an alternative method for deriving Sudakov Minoration type estimates via "discrepancy" bounds. More precisely, we obtain such bounds between the expected suprema of canonical processes generated by random vectors with independent coordinates and the expected suprema of Gaussian processes. In particular, we obtain a refined proximity estimate for Rademacher and Gaussian complexities. Our estimates are dimension-free and depend only on the geometric parameters and the numerical complexity of the underlying index set. In Chapter 3, we focus on concentration of measure phenomenon and study variancesensitive deviation inequalities. In particular, we derive generalizations of variancesensitive concentration inequalities in [PV18] and in [PV19] to include functions that are "nearly" convex. We also derive an important application in the form of Hanson- Wright type inequality.
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Ph. D.