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Limit theory for unbiased and consistent estimators of statistics of random tessellations

Part of: Limit theorems Geometric probability and stochastic geometry Nonparametric inference

Published online by Cambridge University Press:  16 July 2020

Daniela Flimmel Open the ORCID record for Daniela Flimmel [Opens in a new window] ,
Zbyněk Pawlas Open the ORCID record for Zbyněk Pawlas [Opens in a new window]  and
J. E. Yukich
Daniela Flimmel*
Affiliation:
Charles University
Zbyněk Pawlas*
Affiliation:
Charles University
J. E. Yukich*
Affiliation:
Lehigh University
*
*Postal address: Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic. E-mail: [email protected]
**Postal address: Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic. E-mail: [email protected]
***Postal address: Department of Mathematics, Lehigh University, 14 E. Packer Ave, Bethlehem, PA 18015. E-mail: [email protected]
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Abstract

We observe a realization of a stationary weighted Voronoi tessellation of the d-dimensional Euclidean space within a bounded observation window. Given a geometric characteristic of the typical cell, we use the minus-sampling technique to construct an unbiased estimator of the average value of this geometric characteristic. Under mild conditions on the weights of the cells, we establish variance asymptotics and the asymptotic normality of the unbiased estimator as the observation window tends to the whole space. Moreover, weak consistency is shown for this estimator.

Keywords

central limit theoremweighted Voronoi tessellationminus-samplingPoisson point processstabilizationtypical cell

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry 62G05: Estimation
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 679 - 702
Copyright
© Applied Probability Trust 2020

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References

Baddeley, A. J. (1999). Spatial sampling and censoring, In Stochastic Geometry: Likelihood and Computation, eds O. E. Barndorff-Nielsen, W. S. Kendall and M. N. M. Van Lieshout. Chapman and Hall, London, pp. 37–78.Google Scholar
Baryshnikov, Yu. and Yukich, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Prob. 15, 213–253.10.1214/105051604000000594CrossRefGoogle Scholar
Beneš, V. andRataj, J. (2004). Stochastic Geometry: Selected Topics. Kluwer Academic Publishers, Boston.Google Scholar
Błaszczyszyn, B., Yogeshwaran, D. andYukich, J. E. (2019). Limit theory for geometric statistics of point processes having fast decay of correlations. To appear in Ann. Prob.10.1214/18-AOP1273CrossRefGoogle Scholar
Chiu, S. N., Stoyan, D., Kendall, W. S. andMecke, J. (2013). Stochastic Geometry and its Applications, 3rd edn. Wiley, Chichester.10.1002/9781118658222CrossRefGoogle Scholar
Lachièze-Rey, R., Schulte, M. andYukich, J. E. (2019). Normal approximation for stabilizing functionals. Ann. Appl. Prob., 29, 931–993.10.1214/18-AAP1405CrossRefGoogle Scholar
Lautensack, C. andZuyev, S. (2008). Random Laguerre tessellations. Adv. Appl. Prob. 40, 630–650.10.1239/aap/1222868179CrossRefGoogle Scholar
McGivney, K. andYukich, J. E. (1999). Asymptotics for Voronoi tessellations on random samples. Stoch. Process. Appl. 83, 273–288.10.1016/S0304-4149(99)00035-6CrossRefGoogle Scholar
Miles, R. E. (1974). On the elimination of edge-effects in planar sampling. In Stochastic Geometry: A Tribute to the Memory of Rollo Davidson, eds E. F. Harding and D. G. Kendall. John Wiley and Sons, London, pp. 228–247.Google Scholar
Møller, J. (1992). Random Johnson–Mehl tessellations. Adv. Appl. Prob. 24, 814–844.10.2307/1427714CrossRefGoogle Scholar
Okabe, A., Boots, B., Sugihara, K. andChiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. Wiley, Chichester.10.1002/9780470317013CrossRefGoogle Scholar
Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Prob. 12, 989–1035.10.1214/EJP.v12-429CrossRefGoogle Scholar
Penrose, M. D. (2007). Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 1124–1150.10.3150/07-BEJ5167CrossRefGoogle Scholar
Penrose, M. D. andYukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Prob. 11, 1005–1041.CrossRefGoogle Scholar
Penrose, M. D. andYukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.CrossRefGoogle Scholar
Schneider, R. andWeil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.10.1007/978-3-540-78859-1CrossRefGoogle Scholar