Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering

TL;DR

This paper introduces a new inequality on the Poisson space using Malliavin calculus and interpolation, enabling the analysis of mixed Gaussian and Poisson limits, with applications to random geometric graphs.

Contribution

It develops a novel inequality on the Poisson space that unifies and extends previous results, facilitating multidimensional stable convergence and mixed limit theorems.

Findings

Established new criteria for multidimensional stable convergence.

Derived conditions for mixed Gaussian and Poisson limit theorems.

Applied results to joint sub-graph counting in random geometric graphs.

Abstract

Using Malliavin operators together with an interpolation technique inspired by Arratia, Goldstein and Gordon (1989), we prove a new inequality on the Poisson space, allowing one to measure the distance between the laws of a general random vector, and of a target random element composed of Gaussian and Poisson random variables. Several consequences are deduced from this result, in particular: (1) new abstract criteria for multidimensional stable convergence on the Poisson space, (2) a class of mixed limit theorems, involving both Poisson and Gaussian limits, (3) criteria for the asymptotic independence of $U$-statistics obeying to Gaussian and Poisson asymptotic regimes. Our results generalize and unify several previous findings in the field. We provide an application to joint sub-graph counting in random geometric graphs.