Concentration Inequalities for Ultra Log-Concave Distributions

TL;DR

This paper proves concentration inequalities for ultra log-concave distributions, demonstrating they satisfy Poisson bounds and applying these results to intrinsic volumes of convex bodies, improving previous bounds.

Contribution

It introduces new concentration inequalities for ultra log-concave distributions and extends these results to geometric measures, enhancing prior work.

Findings

Ultra log-concave distributions satisfy Poisson concentration bounds.

Derived improved concentration bounds for intrinsic volumes of convex bodies.

Generalized and strengthened previous geometric concentration results.

Abstract

We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an application, we derive concentration bounds for the intrinsic volumes of a convex body, which generalizes and improves a result of Lotz, McCoy, Nourdin, Peccati, and Tropp (2019).