Central limit theorems for U-statistics of Poisson point processes

• Matthias Reitzner ^[2] ; Matthias Schulte ^[1]
  1. [1] Karlsruhe Institute of Technology

     Karlsruhe Institute of Technology

     Stadtkreis Karlsruhe, Alemania

     
  2. [2] University of Osnabrueck
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 6, 2013, págs. 3879-3909
• Idioma: inglés
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• Resumen

  • A U-statistic of a Poisson point process is defined as the sum ∑f(x1,…,xk) over all (possibly infinitely many) k-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener–Itô chaos expansion of such a functional is computed and used to derive a formula for the variance. Central limit theorems for U-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable. As applications, the intersection process of Poisson hyperplanes and the length of a random geometric graph are investigated.