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

Solesne Bourguin (University of Luxembourg)
Giovanni Peccati (University of Luxembourg)

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 following 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.

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Pages: 1-42

Publication Date: August 11, 2014

DOI: 10.1214/EJP.v19-2879

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