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

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

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: daniela.flimmel@karlin.mff.cuni.cz
**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: pawlas@karlin.mff.cuni.cz
***Postal address: Department of Mathematics, Lehigh University, 14 E. Packer Ave, Bethlehem, PA 18015. E-mail: jey0@lehigh.edu
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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

Information

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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