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

Abstract

We study point processes that consist of certain centres 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 Poisson-Delaunay mosaic. As a consequence, we establish scaling limits for maxima of concrete size functionals. This extends extreme value results from (Extremes 17 (2014) 359–385) and (Stochastic Process. Appl. 124 (2014) 2917–2953).