For a given locally finite system of points in Euclidean space, Voronoi tessellation (VT) is a division of the space into polyhedra (into polygons in the case of the plane) “about” the points of the system. Precisely, the Voronoi polygon (cell) about a chosen point of the system is the subset of points of the space that lie closer to the chosen point than to any other point of the system. If the underlying system of points is a Poisson point process, the resulting random tessellation is called the Poisson Voronoi tessellation (PVT). In the present paper PVT is generated by an inhomogeneous Poisson point process whose intensity takes different constant values on sets of some finite partition of the space (some class of modulated-Poisson point processes (mPVT)). In this case, the distribution of “typical cell” of PVT depends on the location and is interpreted as conditional, given that the underlying process has a point at this location (formal definition requires Palm distribution theory, see Section 2).
Known formulae for distributional properties of the typical cell of PVT are almost entirely confined to the homogeneous case. Even then, formulae are very complicated and mainly approximations are known [see the review on Section 10.6 of D. Stoyan, W. Kendall and J. Mecke, “Stochastic geometry and its applications” (1995; Zbl 0838.60002)]. Note that the cell of the VT about a given point is fully shaped by the neighbors of that point in the system of generating points. Thus, provided that the partition of the space is not very “fine” with respect to the intensities of the points, the resulting mPVT is a “locally homogeneous” PVT. Consequently, the “typical cell of a given partitioning set” is highly probably identical to the typical cell of the homogeneous scenario, and a “randomly chosen cell from the whole mPVT” should have a distribution close to the mixture of the homogeneous cases. The authors consider mPVT which is a so-called near-completely decomposable model (more precisely, “near-completely decomposable in mean”).
The paper is organized as follows: In Section 2 the authors introduce mPVT and recall notions and facts concerning point processes and random closed sets. A general modulated marked point process is introduced in Section 3. The main results concerning the approximation decomposition of the mPVT with deterministic and random modulation are given in Sections 4 and 5, respectively.