Ratio limits and simulation algorithms for the Palm version of stationary iterated tessellations

Abstract

Distributional properties and a simulation algorithm for the Palm version of stationary iterated tessellations are considered. In particular, we study the limit behaviour of functionals related to Cox–Voronoi cells (such as typical shortest-path lengths) if either the intensity γ0 of the initial tessellation or the intensity γ1 of the component tessellation converges to 0. We develop an explicit description of the Palm version of Poisson–Delaunay tessellations (PDT), which provides a new direct simulation algorithm for the typical Cox–Voronoi cell based on PDT. It allows us to simulate the Palm version of stationary iterated tessellations, where either the initial or component tessellation is a PDT and can furthermore be used in order to show numerically that the qualitative and quantitative behaviour of certain functionals related to Cox–Voronoi cells strongly depends on the type of the underlying iterated tessellation.