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On the almost sure coverage property of Voronoi tessellation: the ℝ1 case

Part of: Inference from stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Estate Khmaladze  and
N. Toronjadze
Estate Khmaladze*
Affiliation:
University of New South Wales and A. Razmadze Mathematical Institute, Tbilisi
N. Toronjadze*
Affiliation:
University of New South Wales
*
∗ Postal address: Department of Statistics, University of New South Wales, Sydney 2052, Australia.
∗ Postal address: Department of Statistics, University of New South Wales, Sydney 2052, Australia.
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Abstract

This paper raises the following question: let {Φn(A), A ⊂ ℝd} be a Poisson process with intensity nf(x), x ∈ ℝd and let c(Xi | Φn) be a Voronoi tile with nucleus Xi (a jump point of Φn). Let μ(.) denote Lebesgue measure in ℝd. Is it true that, for any bounded measurable subset B of ℝd, ∑Xi∈Bμ(c(Xi| Φn)) → μ(B) almost surely as n → ∞ only if f > 0 almost everywhere? This statement can be viewed as the strong law of large numbers for Voronoi tessellation. Though the positive answer may seem ‘obvious’, we could not find any such statement, especially for arbitrary measurable B and nonhomogeneous Poisson processes. For B with the boundary of Lebesgue measure 0 the proof is simple. We prove in this paper that the statement is true for ℝ1.

Keywords

Poisson point processstrong law of large numbersimage analysisdata driven pixelsnonparametric re-scaling of a distribution to the uniform distributionspacingsmeasure-valued martingalesmartingale limit theorems

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62M40: Random fields; image analysis
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 756 - 764
Copyright
Copyright © Applied Probability Trust 2001 

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