Limits of Poisson-Laguerre tessellations

Abstract

For sequences of Poisson-Laguerre tessellations and their duals in $\mathbb{R}^d$, generated by Poisson point processes $(\eta_n)_{n\in\mathbb{N}}$ in $\mathbb{R}^d \times \mathbb{R}$, we prove limit theorems as $n\to \infty$. The intensity measure of $\eta_n$ has density of the form $(v,h)\mapsto f_n(h)$ with respect to the Lebesgue measure, where $v\in \mathbb{R}^d$ and $h\in \mathbb{R}$. Identifying a tessellation with its skeleton (the union of the boundaries of all its cells) we provide verifiable conditions on $(f_n)_{n\in\mathbb{N}}$ that ensure convergence either to the classical Poisson-Voronoi/Poisson-Delaunay tessellation or to another Poisson-Laguerre tessellation. We also discuss convergence of the corresponding typical cells. As a corollary, we show that the Poisson-Voronoi and the Poisson-Delaunay tessellations arise as limits of the $\beta$-Voronoi and the $\beta$-Delaunay tessellations, respectively, as $\beta\to -1$.