Laws of large numbers in stochastic geometry with statistical applications

TL;DR

This paper establishes laws of large numbers for measures derived from stabilized random vectors in stochastic geometry, with applications to set estimation and variance estimation in nonparametric regression.

Contribution

It introduces general weak and strong laws of large numbers for stabilized measures and applies these to set and variance estimation problems.

Findings

Consistent estimation of an unknown set using Voronoi cells.

Laws of large numbers for stabilized measures in stochastic geometry.

Application to Gamma statistic for variance estimation.

Abstract

Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $\nu_n=\sum_i\xi_i$, where $\xi_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $\xi_i$ is determined by the (suitably rescaled) set of points near $X_i$. For bounded test functions $f$ on $R^d$, we give weak and strong laws of large numbers for $\nu_n(f)$. The general results are applied to demonstrate that an unknown set $A$ in $d$-space can be consistently estimated, given data on which of the points $X_i$ lie in $A$, by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze. Further applications are given concerning the Gamma statistic for estimating the variance in nonparametric regression.