Limits of Poisson-Laguerre tessellations

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 $n\to \mathrm{\infty }$. 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 $\beta \to -1$.