Multivariate Second-Order $p$-Poincar\'e Inequalities

TL;DR

This paper introduces improved bounds for multivariate normal approximation of Poisson functionals, reducing moment assumptions from fourth to nearly second moments, enabling new applications in random geometric graphs.

Contribution

It provides new bounds requiring only (2+ε)-moments for multivariate Poisson functionals, advancing beyond previous fourth-moment requirements.

Findings

Quantitative CLTs for multivariate functionals of random geometric graphs.

Reduced moment assumptions enable analysis of previously intractable models.

Application to Gilbert graphs demonstrates practical impact.

Abstract

In this work, we discuss new bounds for the normal approximation of multivariate Poisson functionals under minimal moment assumptions. Such bounds require one to estimate moments of so-called add-one costs of the functional. Previous works required the estimation of $4^{\text{th}}$ moments, while our result only requires $(2 + \epsilon)$-moments, based on recent improvements introduced by (Trauthwein 2022). As applications, we show quantitative CLTs for two multivariate functionals of the Gilbert, or random geometric, graph. These examples were out of range for previous methods.