Point process convergence of large inradii of Poisson-Laguerre tessellations

Abstract

In this paper we study a weighted generalization of the Poisson-Voronoi tessellation called the Poisson-Laguerre tessellation, where the nuclei of the generating Poisson process additionally carry independent non-negative random weights. For each cell we can define its inradius as the radius of the largest ball centered at the nucleus and contained in the cell. We consider point processes of nuclei, weights and inradii, where the nuclei are taken from growing observation windows and the processes are suitably rescaled and shifted to see the behavior of large inradii. We prove convergence in distribution to suitable Poisson processes, obtaining as corollaries the asymptotic behavior of the maximal inradii. Our results cover dimension $d \geq 3$ and bounded random weights, dimension $d = 2$ and random weights with suitable finite exponential moments as well as dimension $d \geq 2$ and random weights following a power-law distribution. Particularly, we observe a different behavior of the Poisson-Laguerre tessellation in the planar and higher dimensional cases when the weights are bounded. The proofs of our results are based on suitable Poisson process approximation techniques and a careful investigation of the geometry of large cells.