scholarly journals On the local approximation of mean densities of random closed sets

Bernoulli ◽  
2014 ◽  
Vol 20 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Elena Villa
Keyword(s):  
Local Approximation ◽  
Closed Sets ◽  
Random Closed Sets

Random measurable sets and covariogram realizability problems

2015 ◽  
Vol 47 (3) ◽  
pp. 611-639 ◽  
Author(s):  
Bruno Galerne ◽  
Raphael Lachièze-Rey
Keyword(s):  
Stochastic Geometry ◽  
Materials Science ◽  
Local Approximation ◽  
Positive Operators ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Geometric Nature ◽  
The Way

We provide a characterization of realisable set covariograms, bringing a rigorous yet abstract solution to the S2 problem in materials science. Our method is based on the covariogram functional for random measurable sets (RAMS) and on a result about the representation of positive operators on a noncompact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, and they provide a weaker framework that allows the manipulation of more irregular functionals, such as the perimeter. We therefore use the illustration provided by the S2 problem to advocate the use of RAMS for solving theoretical problems of a geometric nature. Along the way, we extend the theory of random measurable sets, and in particular the local approximation of the perimeter by local covariograms.

Random measurable sets and covariogram realizability problems

2015 ◽  
Vol 47 (03) ◽  
pp. 611-639 ◽  
Author(s):  
Bruno Galerne ◽  
Raphael Lachièze-Rey
Keyword(s):  
Stochastic Geometry ◽  
Materials Science ◽  
Local Approximation ◽  
Positive Operators ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Geometric Nature ◽  
The Way

We provide a characterization of realisable set covariograms, bringing a rigorous yet abstract solution to theS2problem in materials science. Our method is based on the covariogram functional for random measurable sets (RAMS) and on a result about the representation of positive operators on a noncompact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, and they provide a weaker framework that allows the manipulation of more irregular functionals, such as the perimeter. We therefore use the illustration provided by theS2problem to advocate the use of RAMS for solving theoretical problems of a geometric nature. Along the way, we extend the theory of random measurable sets, and in particular the local approximation of the perimeter by local covariograms.

scholarly journals An elementary renewal theorem for random compact convex sets

1995 ◽  
Vol 27 (4) ◽  
pp. 931-942 ◽  
Author(s):  
Ilya S. Molchanov ◽  
Edward Omey ◽  
Eugene Kozarovitzky
Keyword(s):  
Support Function ◽  
Convex Sets ◽  
Compact Convex ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Minkowski Sums ◽  
Image Position ◽  
Unit Vectors

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

scholarly journals Poisson hail on a hot ground

2011 ◽  
Vol 48 (A) ◽  
pp. 343-366
Author(s):  
Francois Baccelli ◽  
Sergey Foss
Keyword(s):  
Euclidean Space ◽  
Service Time ◽  
Evolution Equations ◽  
Stationary Regime ◽  
Service Discipline ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Exponential Moments ◽  
First In First Out ◽  
Arrival Intensity

We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACSs) of the Euclidean space. These RACSs arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACSs can be served simultaneously and service is in the first-in–first-out order, i.e. only the hailstones in contact with the ground melt at speed 1, whereas the others are queued. A tagged RACS waits until all RACSs that arrived before it and intersecting it have fully melted before starting its own melting. We give the evolution equations for this queue. We prove that it is stable for a sufficiently small arrival intensity, provided that the typical diameter of the RACS and the typical service time have finite exponential moments. We also discuss the percolation properties of the stationary regime of the RACS in the queue.

Size–biased random closed sets

1999 ◽  
Vol 32 (9) ◽  
pp. 1631-1644 ◽  
Author(s):  
M.N.M. van Lieshout
Keyword(s):  
Closed Sets ◽  
Random Closed Sets
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