Final Oral Public Examination

Vector-valued concentration inequalities on discrete spaces

Advisor: Ramon van Handel

Abstract:

Existing concentration inequalities for functions that take values in a general Banach space, such as the classical results of Pisier (1986) and the more recent contribution of Ivanisvili, van Handel, and Volberg (2020), are known only in very special settings, such as the Gaussian measure on Rn and the uniform measure on the discrete cube {−1, 1}^n.

In this thesis, we prove such vector-valued concentration inequalities for more general probability measures on discrete spaces. In the first chapter, we present such an inequality for the biased product measure on the discrete cube with an optimal dependence on the  bias parameter and the Rademacher type of the target Banach space. This result yields scaling limits to the product of Poisson measures, as well as lower bounds on the average distortion of embeddings of the discrete cube into Banach spaces of nontrivial type, implying average nonembeddability.

Moreover, we present a novel vector-valued concentration inequality for the uniform measure on the symmetric group, which is the first to go beyond the setting of product measures. Our inequality attains optimal dimensional dependency for Banach space of Rademacher type p ∈ [1, 2), which implies average nonembeddability of the symmetric group into Banach spaces of nontrivial Rademacher type. Our approach enriches the Markov semigroup interpolation argument used by Ivanisvili, van Handel, and Volberg. In particular, we further techniques for capturing the concentration of random coefficients arising from the semigroup method.