## Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering.(English)Zbl 1316.60089

Summary: Using Malliavin operators together with an interpolation technique inspired by [R. Arratia et al., Ann. Probab. 17, No. 1, 9–25 (1989; Zbl 0675.60017)], 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.

### MSC:

 60H07 Stochastic calculus of variations and the Malliavin calculus 60F05 Central limit and other weak theorems 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G15 Gaussian processes 60D05 Geometric probability and stochastic geometry 05C80 Random graphs (graph-theoretic aspects)

Zbl 0675.60017
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