Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry

• Decreusefond, Laurent ^[3] ; Schulte, Matthias ^[1] ; Thäle, Christoph ^[2]
  1. [1] Karlsruhe Institute of Technology

     Karlsruhe Institute of Technology

     Stadtkreis Karlsruhe, Alemania

     
  2. [2] Ruhr University Bochum

     Ruhr University Bochum

     Kreisfreie Stadt Bochum, Alemania

     
  3. [3] Telecom ParisTech

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 3, 2016, págs. 2147-2197
• Idioma: inglés
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• Resumen

  • A Poisson or a binomial process on an abstract state space and a symmetric function f acting on k-tuples of its points are considered. They induce a point process on the target space of f. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein’s method for Poisson process approximation, a Glauber dynamics representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived, and examples from stochastic geometry are investigated.