Probability and Statistics Seminar: Vector concentration inequalities

Friday, April 16 at 3:30pm to 4:30pm

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Ramon van Handel, Princeton University

Abstract: How do classical concentration inequalities extend to functions taking values in normed spaces? Such questions arise in various settings (in functional analysis, random matrix theory, etc.), and are generally studied on a case by case basis. In the Gaussian case, however, Pisier discovered a powerful principle that explains in complete generality how vector-valued functions concentrate. Unfortunately, Pisier's argument appears to be very specific to the Gaussian distribution, and until recently this phenomenon was not known to hold in any other setting.

In this talk, I will explain that Pisier's principle is much more universal than expected: a completely analogous result holds under the classical (Bakry-Emery) conditions used in the theory of scalar concentration inequalities. Moreover, a modified form of this principle holds also in discrete settings (as was first observed, in the special case of the discrete cube, in joint work with Ivanisvili and Volberg that settled an old conjecture of Enflo in Banach space theory). These new inequalities provide very general tools for establishing concentration properties of vector-valued functions.

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