Weak laws of large numbers in geometric probability
type: Item
instance of: scholarly article
title: Weak laws of large numbers in geometric probability
zbMATH Open document ID: 1029.60008
author: Mathew D. Penrose
author: J. E. Yukich
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: The authors use a coupling argument to establish a general weak law of large numbers for functionals of binomial point processes in \(d\)-dimensional Euclidean space. The limit depends explicitly on the density, possibly non-uniform, of the process. When the point set forms the vertex set of a graph, functionals of interest include total edge length with arbitrary weighting, number of edges and number of components. The general result is applied to various graphs. Weak laws of large numbers are also obtained for functionals of marked point processes, including statistics of Boolean models. The methods were used previously to obtain a strong law of large numbers for functionals on uniform point sets; see e.g. the authors [Ann. Appl. Probab. 12, 272-301 (2002; Zbl 1018.60023)].
reviewed by: M. P. Quine
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 68U05
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1906136
zbMATH Keywords: weak law of large numbers
zbMATH Keywords: computational geometry
zbMATH Keywords: objective method
zbMATH Keywords: minimal spanning tree
zbMATH Keywords: nearest neighbours graph
zbMATH Keywords: Voronoi graph
zbMATH Keywords: sphere of influence graph
zbMATH Keywords: proximity graph
zbMATH Keywords: Boolean model
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1042765669
OpenAlex ID: W2021695397
cites work: Asymptotics for Euclidean minimal spanning trees on random points
cites work: Simultaneous uniqueness of infinite clusters in stationary random labeled graphs
cites work: Percolation and minimal spanning trees
cites work: Modern multidimensional scaling: theory and applications
cites work: Graph-theoretic procedures for dimension identification
cites work: The expected size of some graphs in computational geometry
cites work: On nearest-neighbor graphs
cites work: Q4352325
cites work: Q4693063
cites work: Q4040734
cites work: The proportion of triangles in a Poisson-Voronoi tessellation of the plane
cites work: On the fraction of random points by specified nearest-neighbour interrelations and degree of attraction
cites work: On the variance of the random sphere of influence graph
cites work: Strong laws for Euclidean graphs with general edge weights
cites work: The central limit theorem for weighted minimal spanning trees on random points
cites work: Q4084463
cites work: Asymptotics for Voronoi tessellations on random samples
cites work: Statistics of the Boolean model for practitioners and mathematicians
cites work: The random minimal spanning tree in high dimensions
cites work: Random parking, sequential adsorption, and the jamming limit
cites work: Central limit theorems for some graphs in computational geometry.
cites work: Limit theory for random sequential packing and deposition
cites work: Q5691080
cites work: On the number of leaves of a euclidean minimal spanning tree
cites work: Probability theory of classical Euclidean optimization problems
cites work: Asymptotics for weighted minimal spanning trees on random points
DOI: 10.1214/AOAP/1042765669
description: scientific article; zbMATH DE number 1906136