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

Solesne Bourguin (1) , Giovanni Peccati (1)

1 FSTC - Faculté des Sciences, de la Technologie et de la Communication

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.

Mots clés: Chen-Stein Method; Contractions; Malliavin Calculus; Poisson Limit Theorems; Poisson Space; Random Graphs; Total Variation Distance; Wiener Chaos

Domaines: Probabilités [math.PR]

Solesne Bourguin, Giovanni Peccati. Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering. 2012. ⟨hal-00731861v2⟩