Poisson approximation of Poisson-driven point processes and extreme values in stochastic geometry

Abstract: We study point processes that consist of certain centers of point tuples of an underlying Poisson process. Such processes arise in stochastic geometry in the study of exceedances of various functionals describing geometric properties of the Poisson process. We use a coupling of the point process with its Palm version to prove a general Poisson limit theorem. We then combine our general result with the theory of asymptotic shapes of large cells (Kendall's problem) in random mosaics and prove Poisson limit theorems for large cells (with respect to a general size functional) in the Poisson-Voronoi and -Delaunay mosaic. As a consequence, we establish Gumbel limits for the asymptotic distribution of concrete size functionals and specify the rate of convergence. This extends extreme value results from Calka and Chenavier (2014) and Chenavier (2014).