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

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.