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

Published online by Cambridge University Press:  16 July 2020

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

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.

Type
Research Papers
Copyright
© Applied Probability Trust 2020

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