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We study point processes that consist of certain centers of point tuples ofan underlying Poisson process. Such processes arise in stochastic geometry inthe study of exceedances of various functionals describing geometric propertiesof the Poisson process. We use a coupling of the point process with its Palmversion to prove a general Poisson limit theorem. We then combine our generalresult with the theory of asymptotic shapes of large cells (Kendall's problem)in random mosaics and prove Poisson limit theorems for large cells (withrespect to a general size functional) in the Poisson-Voronoi and -Delaunaymosaic. As a consequence, we establish Gumbel limits for the asymptoticdistribution of concrete size functionals and specify the rate of convergence.This extends extreme value results from Calka and Chenavier (2014) andChenavier (2014).

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