Stochastic Analysis for Poisson Point Processes

Volume 7 of the series Bocconi & Springer Series pp 295-310


U-Statistics on the Spherical Poisson Space

  • Solesne BourguinAffiliated withDepartment of Mathematics and Statistics, Boston University Email author 
  • , Claudio DurastantiAffiliated withFaculty of Mathematics, Ruhr University Bochum
  • , Domenico MarinucciAffiliated withDepartment of Mathematics, University of Rome Tor Vergata
  • , Giovanni PeccatiAffiliated withUnité de Recherche en Mathématiques, Université du Luxembourg

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We review a recent stream of research on normal approximations for linear functionals and more general U-statistics of wavelets/needlets coefficients evaluated on a homogeneous spherical Poisson field. We show how, by exploiting results from Peccati and Zheng (Electron J Probab 15(48):1487–1527, 2010) based on Malliavin calculus and Stein’s method, it is possible to assess the rate of convergence to Gaussianity for a triangular array of statistics with growing dimensions. These results can be exploited in a number of statistical applications, such as spherical density estimations, searching for point sources, estimation of variance, and the spherical two-sample problem.