Vector-valued concentration inequalities on the biased discrete cube

TL;DR

This paper develops vector-valued concentration inequalities for biased discrete hypercubes, improving dependence on bias and Banach space properties, with applications to measure concentration and embedding distortion bounds.

Contribution

It introduces optimal vector-valued concentration inequalities for biased measures on the hypercube, extending to Poisson product measures and establishing lower bounds on embedding distortions.

Findings

Optimal dependence on bias parameter and Banach space type achieved.

New concentration inequalities for Poisson product measures derived.

Lower bounds on hypercube embedding distortions established.

Abstract

We present vector-valued concentration inequalities for the biased measure on the discrete hypercube with an optimal dependence on the bias parameter and the Rademacher type of the target Banach space. These results allow us to obtain novel vector-valued concentration inequalities for the measure given by a product of Poisson distributions. Furthermore, we obtain lower bounds on the average distortion with respect to the biased measure of embeddings of the hypercube into Banach spaces of nontrivial type which imply average non-embeddability.