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

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.

References in corpus (10)

• Central limit theorems for $U$-statistics of Poisson point processes
• Central limit theorems for Poisson hyperplane tessellations
• Central limit theorems for double Poisson integrals
• The Chen-Stein method for Poisson functionals
• Fourth Moment Theorem and q-Brownian Chaos
• Malliavin-Stein method for multi-dimensional U-statistics of Poisson point processes
• Exact and asymptotic results for intrinsic volumes of Poisson k-flat processes
• The scaling limit of Poisson-driven order statistics with applications in geometric probability
• A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process
• A Central Limit Theorem for the Poisson-Voronoi Approximation