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

Abstract

We observe a realization of a stationary weighted Voronoi tessellation of thed-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 theorem
• weighted Voronoi tessellation
• minus-sampling
• Poisson point process
• stabilization
• typical cell