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

## TL;DR

This paper develops unbiased, consistent, and asymptotically normal estimators for geometric features of cells in stationary weighted Voronoi tessellations, with theoretical variance and distribution results as observation windows expand.

## Contribution

It introduces a minus-sampling based unbiased estimator for cell characteristics and proves its asymptotic properties under mild conditions.

## Key findings

- Estimator is unbiased and weakly consistent.
- Variance asymptotics are established.
- Asymptotic normality is proven.

## Abstract

We observe a realization of a stationary generalized 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, the weak consistency is shown for this estimator.