Noncommutative and vector-valued Rosenthal inequalities

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This thesis is dedicated to the study of a class of probabilistic inequalities, called Rosenthal inequalities. These inequalities provide two-sided estimates for the p-th moments of the sum of a sequence of independent, mean zero random variables in terms of a suitable norm on the sequence itself. Rosenthal inequalities are named after H.P. Rosenthal, who first discovered them for scalar-valued random variables around 1970. The main results of this thesis extend Rosenthal's inequalities in two different directions. In the first part we consider random variables taking values in a Banach space. The main results give Rosenthal-type inequalities in the case where the Banach space is either a Hilbert space or an Lp-space. The inequalities developed in this setting are principally designed to prove a novel Ito isomorphism for vector-valued stochastic integrals with respect to a compensated Poisson random measure. These kind of isomorphisms are a key tool in the analysis of stochastic partial differential equations. The Rosenthal-type inequalities are further extended to apply to random variables taking values in a noncommutative Lp-space associated with a von Neumann algebra. By specializing this result to von Neumann algebras of square matrices, quantitative bounds are found for the moments of the largest singular value of a random matrix in terms of its entries. In the second part of this thesis Rosenthal's original inequalities are generalized to sequences of noncommutative random variables, given by elements of a noncommutative symmetric space. As is the case in the first part, these noncommutative Rosenthal inequalities are applied to obtain Ito isomorphisms for stochastic integrals. For the proof of the noncommutative Rosenthal inequalities several new tools are developed which are interesting in their own right. Novel results are found for other probabilistic inequalities in noncommutative symmetric spaces, such as Khintchine and Burkholder-Gundy inequalities, as well as results in the interpolation theory for such spaces.