Typical Poisson-Voronoi Cell Research Articles

Concentration and moderate deviations for Poisson polytopes and polyhedra

The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to a class of zero cells associated with Poisson hyperplane mosaics. As a special case, this comprises the typical Poisson–Voronoi cell conditioned on having large inradius.

Asymptotic Blind-Spot Analysis of Localization Networks Under Correlated Blocking Using a Poisson Line Process

In a localization network, the line-of-sight between anchors (transceivers) and targets may be blocked due to the presence of obstacles in the environment. Due to the non-zero size of the obstacles, the blocking is typically correlated across both anchor and target locations, with the extent of correlation increasing with obstacle size. If a target does not have line-of-sight to a minimum number of anchors, then its position cannot be estimated unambiguously and is, therefore, said to be in a blind-spot . However, the analysis of the blind-spot probability of a given target is challenging due to the inherent randomness in the obstacle locations and sizes. In this letter, we develop a new framework to analyze the worst-case impact of correlated blocking on the blind-spot probability of a typical target; in particular, we model the obstacles by a Poisson line process and the anchor locations by a Poisson point process. For this setup, we define the notion of the asymptotic blind-spot probability of the typical target and derive a closed-form expression for it as a function of the area distribution of a typical Poisson-Voronoi cell. As an upper bound for the more realistic case when obstacles have finite dimensions, the asymptotic blind-spot probability is useful as a design tool to ensure that the blind-spot probability of a typical target does not exceed a desired threshold, $\boldsymbol {\epsilon }$ .

Asymptotic properties of Euclidean shortest-path trees in random geometric graphs

We consider asymptotic properties of two functionals on Euclidean shortest-path trees appearing in random geometric graphs in R2 which can be used, for example, as models for fixed-access telecommunication networks. First, we determine the asymptotic bivariate distribution of the two backbone lengths inside a certain class of typical Cox–Voronoi cells as the size of this cell grows unboundedly. The corresponding Voronoi tessellation is generated by a stationary Cox process which is concentrated on the edges of the random geometric graph and whose intensity tends to 0. The limiting random vector can be represented as a simple geometric functional of a decomposition of a typical Poisson–Voronoi cell induced by an independent random sector. Using similar methods, we consider the asymptotic bivariate distribution of the total lengths of the two subtrees inside the Cox–Voronoi cell.

Random Line Tessellations of the Plane: Statistical Properties of Many-Sided Cells

We consider a family of random line tessellations of the Euclidean plane introduced in a much more formal context by Hug and Schneider [Geom. Funct. Anal. 17, 156 (2007)] and described by a parameter \alpha\geq 1. For \alpha=1 the zero-cell (that is, the cell containing the origin) coincides with the Crofton cell of a Poisson line tessellation, and for \alpha=2 it coincides with the typical Poisson-Voronoi cell. Let p_n(\alpha) be the probability for the zero-cell to have n sides. By the methods of statistical mechanics we construct the asymptotic expansion of \log p_n(\alpha) up to terms that vanish as n\to\infty. In the large-n limit the cell is shown to become circular. The circle is centered at the origin when \alpha>1, but gets delocalized for the Crofton cell, \alpha=1, which is a singular point of the parameter range. The large-n expansion of \log p_n(1) is therefore different from that of the general case and we show how to carry it out. As a corollary we obtain the analogous expansion for the {\it typical} n-sided cell of a Poisson line tessellation.

Second‐order properties of the point process of nodes in a stationary Voronoi tessellation

AbstractIn this paper we derive representation formulae for the second factorial moment measure of the point process of nodes and the second moment of the number of vertices of the typical cell associated with a stationary normal Voronoi tessellation in ℝd . In case the Voronoi tessellation is generated by a stationary Poisson process with intensity λ > 0 the corresponding pair correlation function gV,λ (r) can be expressed by a weighted sum of d +2 (numerically tractable) multiple parameter integrals. The asymptotic variance of the number of nodes in an increasing cubic domain as well as the second moment of the number of vertices of the typical Poisson Voronoi cell are calculated exactly by means of these parameter integrals. The existence of a (d − 1)st‐order pole of gV,λ (r) at r = 0 is proved and the exact value of limr →0 rd –1 gV,λ (r) is determined. In the particular cases d = 2 and d = 3 the graph of gV,1(r) including its local extreme points, the points of level 1 of gV, 1(r) and other characteristics are computed by numerical integration. Furthermore, an asymptotically exact confidence interval for the intensity of nodes is obtained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the correlation between the volumes of the typical Poisson-Voronoi cell and the typical Stienen sphere

In this paper we consider a tessellation V generated by a homogeneous Poisson process Φ in R d and, furthermore, the random set of spheres with centres being the points in Φ and having radii equal to half the distance to their closest neighbouring point in Φ. In R d we give an integral formula for the correlation between the volume of the typical cell and the volume of the sphere in the typical cell, and we also show that this correlation is strictly positive. Furthermore, on the real line we give an analytical expression for the correlation, and in the plane and in space we give simplified integral formulae. Numerical values for the correlation for d = 2,…,7 are also given.

On the correlation between the volumes of the typical Poisson-Voronoi cell and the typical Stienen sphere

In this paper we consider a tessellation V generated by a homogeneous Poisson process Φ in Rd and, furthermore, the random set of spheres with centres being the points in Φ and having radii equal to half the distance to their closest neighbouring point in Φ. In Rd we give an integral formula for the correlation between the volume of the typical cell and the volume of the sphere in the typical cell, and we also show that this correlation is strictly positive. Furthermore, on the real line we give an analytical expression for the correlation, and in the plane and in space we give simplified integral formulae. Numerical values for the correlation for d = 2,…,7 are also given.

Limit theorems for the typical Poisson–Voronoi cell and the Crofton cell with a large inradius

In this paper, we are interested in the behavior of the typical Poisson–Voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity. We prove a law of large numbers for its number of vertices and the area of the cell outside the disk. Moreover, for the latter, we establish a central limit theorem as well as moderate deviation type results. The proofs deeply rely on precise connections between Poisson–Voronoi tessellations, convex hulls of Poisson samples and germ–grain models in the unit ball. Besides, we derive analogous facts for the Crofton cell of a stationary Poisson line process in the plane.

An explicit expression for the distribution of the number of sides of the typical Poisson-Voronoi cell

In this paper, we give an explicit expression for the distribution of the number of sides (or equivalently vertices) of the typical cell of a two-dimensional Poisson-Voronoi tessellation. We use this formula to give a table of numerical values of the distribution function.

An explicit expression for the distribution of the number of sides of the typical Poisson-Voronoi cell

In this paper, we give an explicit expression for the distribution of the number of sides (or equivalently vertices) of the typical cell of a two-dimensional Poisson-Voronoi tessellation. We use this formula to give a table of numerical values of the distribution function.