Central limit theorems for Poisson hyperplane tessellations

We derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in $\mathbb{R}^d$. This result generalizes an earlier one proved by Paroux [Adv. in Appl. Probab. 30 (1998) 640--656] for intersection points of motion-invariant Poisson line processes in $\mathbb{R}^2$. Our proof is based on Hoeffding's decomposition of $U$-statistics which seems to be more efficient and adequate to tackle the higher-dimensional case than the ``method of moments'' used in [Adv. in Appl. Probab. 30 (1998) 640--656] to treat the case $d=2$. Moreover, we extend our central limit theorem in several directions. First we consider $k$-flat processes induced by Poisson hyperplane processes in $\mathbb{R}^d$ for $0\le k\le d-1$. Second we derive (asymptotic) confidence intervals for the intensities of these $k$-flat processes and, third, we prove multivariate central limit theorems for the $d$-dimensional joint vectors of numbers of $k$-flats and their $k$-volumes, respectively, in an increasing spherical region.Comment: Published at http://dx.doi.org/10.1214/105051606000000033 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Continuum Line-of-Sight Percolation on Poisson-Voronoi Tessellations

In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson-Voronoi tessellation in the $d$-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition $Z$ of these two processes, two points of $Z$ are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0-1 law, a subcritical phase as well as a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.Comment: 30 pages, 4 figures. Accepted for publication in Advances in Applied Probabilit

Continuum percolation for Cox point processes

We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub- and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in large-radius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.Comment: 21 pages, 5 figure

Percolation for D2D networks on street systems

We study fundamental characteristics for the connectivity of multi-hop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as Poisson-Voronoi or Poisson-Delaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical device-intensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold

Limit theorems for functionals on the facets of stationary random tessellations

We observe stationary random tessellations $X=\{\Xi_n\}_{n\ge1}$ in $\mathbb{R}^d$ through a convex sampling window $W$ that expands unboundedly and we determine the total $(k-1)$-volume of those $(k-1)$-dimensional manifold processes which are induced on the $k$-facets of $X$ ($1\le k\le d-1$) by their intersections with the $(d-1)$-facets of independent and identically distributed motion-invariant tessellations $X_n$ generated within each cell $\Xi_n$ of $X$. The cases of $X$ being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In both cases, however, we obtain that all of the total volumes measured in $W$ are approximately normally distributed when $W$ is sufficiently large. Structural formulae for mean values and asymptotic variances are derived and explicit numerical values are given for planar Poisson--Voronoi tessellations (PVTs) and Poisson line tessellations (PLTs).Comment: Published at http://dx.doi.org/10.3150/07-BEJ6131 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Shortest Path Distance in Manhattan Poisson Line Cox Process

While the Euclidean distance characteristics of the Poisson line Cox process (PLCP) have been investigated in the literature, the analytical characterization of the path distances is still an open problem. In this paper, we solve this problem for the stationary Manhattan Poisson line Cox process (MPLCP), which is a variant of the PLCP. Specifically, we derive the exact cumulative distribution function (CDF) for the length of the shortest path to the nearest point of the MPLCP in the sense of path distance measured from two reference points: (i) the typical intersection of the Manhattan Poisson line process (MPLP), and (ii) the typical point of the MPLCP. We also discuss the application of these results in infrastructure planning, wireless communication, and transportation networks