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 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.

Poisson process approximation under stabilization and Palm coupling

“…The processes we study are functionals of Poisson (or binomial) point processes, which are themselves not Poisson, and in particular lack spatial independence. The results we present here significantly generalize recent ones [15,27] by either considering a stronger approximation distance or more general functionals. Our approach is based on Stein's method for Poisson process approximation; see e.g.…”

Criteria for Poisson process convergence with applications to inhomogeneous Poisson-Voronoi tessellations

“…Using Stein's method, one can even derive quantitative bounds for the Poisson process approximation; see e.g. [1,3,4,5,9,10,13,25,29,30] and the references therein. In contrast to these results, our findings are purely qualitative and do not provide rates of convergence, but they have the advantage that the underlying conditions are easy to verify.…”

Large deviation principle for geometric and topological functionals and associated point processes

“…A variety of functionals in stochastic geometry can be considered as such statistics; see [5,8,25,21]. Moreover, with an appropriate choice of H, the statistics (1.5) can be used to investigate the behavior of topological invariants of a geometric complex [6,7,19,29,37,26].…”

“…With an appropriate choice of scaling regimes, the large deviations of point processes as in (1.4) have been studied by Sanov's theorem and its variant [15,36,16]. In recent times, the process (1.4) also found applications in stochastic geometry [12,25,8]. In particular, the authors of [8] explored the Poisson process approximation of (1.4) by deriving the rate of convergence in terms of the Kantorovich-Rubinstein distance.…”