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

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

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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References

Alves, J. H. G. M and Young, I. R. (2001). Extreme significant wave heights from combined satellite altimeter data. To appear in Proc. 27th ICCE, Sydney, Australia, 2000. In Coastal Engineering 2000 (Proc. 27th Internat. Conf. Coastal Eng., Sydney, 16–21 July 2000), ed. Edge, B. L., American Society of Civil Engineers, New York.Google Scholar
Borovikov, V. P. (1986). Limit theorems for statistics that are partial sums of functions of spacings. Theory Prob. Appl. 32, 86–97.Google Scholar
Copson, E. T. (1965). Asymptotic Expansions. Cambridge University Press.CrossRefGoogle Scholar
Cowan, R. (1980). Properties of ergodic random mosaic processes. Math. Nachr. 97, 89–102.CrossRefGoogle Scholar
Einmahl, J. H. J. and Khmaladze, E. V. (2001). The two-sample problem in Rm and measure-valued martingales. Res. Rep. S98–2, University of New South Wales. In State of the Art in Probability and Statistics (Inst. Math. Statist. Lecture Notes Monog. Ser. 36), eds de Gunst, M., Klaassen, C. and van der Vaart, A., Institute of Mathematical Statistics, Beachwood, OH, pp. 434–463.Google Scholar
Kuroda, K. and Tanemura, H. (1992). Limit theorem and large deviation principle for the Voronoi tessellation generated by a Gibbs point process. Adv. Appl. Prob. 24, 45–70.Google Scholar
Mecke, J. and Muche, L. (1995). The Poisson Voronoi tessellation. I. A basic identity.. 176, 199–208.Google Scholar
Möller, J., (1989). Random tessellations in Rd . Adv. Appl. Prob. 21, 37–73.Google Scholar
Möller, J., (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.Google Scholar
Muche, L. (1996). The Poisson Voronoi tessellations. II. Edge length distribution functions. Math. Nachr. 178, 271–283.Google Scholar
Muche, L. (1998). The Poisson Voronoi tessellation. III. Miles' formula Math. Nachr. 191, 247–267.CrossRefGoogle Scholar
Okabe, A., Boots, B. and Sugihara, K. (1992). Spatial Tessellations Concepts and Applications of Voronoi Diagrams. John Wiley, New York.Google Scholar
Shiryaev, A. N. (1996). Probability, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes With Applications to Statistics. John Wiley, New York.Google Scholar
Stoyan, D. (1986). On generalized planar random tessellations. Math. Nachr. 128, 215–219.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. Akademie, Berlin.Google Scholar
Weiss, V. and Zähle, M. (1988). Geometric measures for random curved mosaics of Rd . Math. Nachr. 138, 313–326.Google Scholar
Zähle, M., (1988). Random cell complexes and generalised sets. Ann. Prob. 16, 1742–1766.Google Scholar