Advanced Tools from Probability Theory

Abstract

This chapter is dedicated to advanced probability results required in the more involved arguments on random measurement matrices, notably precise bounds for Gaussian random matrices and the analysis of random partial Fourier matrices. First, norms of Gaussian vectors in expectation are discussed, followed by Rademacher sums and the symmetrization principle. Khintchine inequalities provide bounds for moments of Rademacher sums. Then decoupling inequalities are covered. They allow one to simplify the analysis of double sums of random variables by replacing one instance of a random vector by an independent copy. The noncommutative Bernstein inequality treated next bounds the tail of a sum of independent random matrices in the operator norm. Dudley’s inequality bounds the supremum of a subgaussian process by an integral over covering numbers with respect to the index set of the process. Slepian’s and Gordon’s lemma compare certain functions of Gaussian vectors in terms of their covariance structures. Together with the concentration of measure principle for Lipschitz functions of Gaussian vectors covered next, they provide powerful tools for the analysis of Gaussian random matrices. Finally, the chapter discusses Talagrand’s inequality, i.e., a Bernstein-type inequality for suprema of empirical processes.