On the distribution of the spherical contact vector of stationary germ-grain models

@article{Last1998OnTD,
  title={On the distribution of the spherical contact vector of stationary germ-grain models},
  author={Guenter Last and Rolf Schassberger},
  journal={Advances in Applied Probability},
  year={1998},
  volume={30},
  pages={36 - 52}
}
We consider a stationary germ-grain model Ξ with convex and compact grains and the distance r(x) from x ε ℝ d to Ξ. For almost all points x ε ℝ d there exists a unique point p(x) in the boundary of Ξ such that r(x) is the length of the vector x-p(x), which is called the spherical contact vector at x. In this paper we relate the distribution of the spherical contact vector to the times it takes a typical boundary point of Ξ to hit another grain if all grains start growing at the same time and at… 
View on Cambridge Press
16 Citations
Highly Influential Citations
2
Background Citations
7
Methods Citations
2
Results Citations
3

16 Citations

On the empty space function of some germ-grain models
On the second derivative of the spherical contact distribution function of smooth grain models
Abstract. We consider a stationary grain model Ξ in ℝd with convex, compact and smoothly bounded grains. We study the spherical contact distribution function F of Ξ and derive (under suitable…
Mean Densities and Spherical Contact Distribution Function of Inhomogeneous Boolean Models
The mean density of a random closed set Θ in ℝ d with Hausdorff dimension n is the Radon–Nikodym derivative of the expected measure 𝔼[ℋ n (Θ ∩·)] induced by Θ with respect to the usual d-dimensional…
On stationary stochastic flows and Palm probabilities of surface processes
We consider a random surface in d tessellating the space into cells and a random vector field u which is smooth on each cell but may jump on . Assuming the pair u stationary we prove a relationship…
On support measures in Minkowski spaces and contact distributions in stochastic geometry
This paper is concerned with contact distribution functions of a random closed set Ξ = ∪∞ n=1 Ξ n in R d , where the Ξ n are assumed to be random nonempty convex bodies. These distribution functions…
GENERALIZED CONTACT DISTRIBUTIONS OF
The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L C Rd and a…
Generalized contact distributions of inhomogeneous Boolean models
The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝ d and a…
Comparisons and asymptotics for empty space hazard functions of germ-grain models
We study stochastic properties of the empty space for stationary germ-grain models in R d ; in particular, we deal with the inner radius of the empty space with respect to a general structuring…
Multiple‐point hit distribution functions and vague convergence of related measures
For a stationary and isotropic random closed set Z in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^d$\end{document} it is a well‐known fact that its…
On Second-Order Characteristics of Germ-Grain Models with Convex Grains
...
...

References

SHOWING 1-10 OF 37 REFERENCES
Central limit theorem for a class of random measures associated with germ-grain models
The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of…
A flow conservation law for surface processes
The object studied in this paper is a pair (Φ, Y), where Φ is a random surface in and Y a random vector field on . The pair is jointly stationary, i.e. its distribution is invariant under…
CHAPTER 5.2 – Stochastic Geometry
Densities for stationary random sets and point processes
For certain stationary random sets X, densities D φ (X) of additive functionals φ are defined and formulas for are derived when K is a compact convex set in . In particular, for the quermassintegrals…
Contact and chord length distribution of a stationary Voronoi tessellation
  • L. Heinrich
  • Mathematics
    Advances in Applied Probability
  • 1998
We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝ d in terms of the Palm void probabilities of the generating…
CHAPTER 5.1 – Integral Geometry
Curvature measures and random sets II
In choosing models of stochastic geometry three general problems play a role which are closely connected with each other: In the present paper second order local geometric properties of random…
Kaplan-Meier estimators of distance distributions for spatial point processes
When a spatial point process is observed through a bounded window, edge effects hamper the estimation of characteristics such as the empty space function F, the nearest neighbor distance distribution…
Kaplan-Meier estimators of interpoint distance distributions for spatial point processes
When a spatial point process is observed through a bounded window, edge eeects hamper the estimation of characteristics such as the empty space function F , the nearest neighbour distance…
Contact and chord length distributions of the Poisson Voronoi tessellation
This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact…
...
...