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

20 May 2020  ·  Otto Moritz ·

We study point processes that consist of certain centers of point tuples of an underlying Poisson process. Such processes can be used in stochastic geometry to study exceedances of various functionals describing geometric properties of the Poisson process... Using a coupling of the point process with its Palm version we prove a general Poisson limit theorem. We then apply our theorem to find the asymptotic distribution of the maximal volume content of random $k$-nearest neighbor balls. Combining our general result with the theory of asymptotic shapes of large cells in random mosaics, we prove a Poisson limit theorem for cell centers in the Poisson-Voronoi and -Delaunay mosaic. As a consequence, we establish Gumbel limits for the asymptotic distribution of the maximal cell size in the Poisson-Voronoi and -Delaunay mosaic w.r.t. a general size functional. read more

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