## Electronic Journal of Probability

### 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 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.

#### Article information

Source
Electron. J. Probab. Volume 19 (2014), paper no. 66, 42 pp.

Dates
Accepted: 11 August 2014
First available in Project Euclid: 4 June 2016

http://projecteuclid.org/euclid.ejp/1465065708

Digital Object Identifier
doi:10.1214/EJP.v19-2879

Mathematical Reviews number (MathSciNet)
MR3248195

Zentralblatt MATH identifier
1316.60089

Rights

#### Citation

Bourguin, Solesne; Peccati, Giovanni. Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering. Electron. J. Probab. 19 (2014), paper no. 66, 42 pp. doi:10.1214/EJP.v19-2879. http://projecteuclid.org/euclid.ejp/1465065708.

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