Laws of large numbers in stochastic geometry with statistical applications

Abstract

Given n independent random marked d-vectors (points) X_i distributed with a common density, define the measure ν_n=∑_iξ_i, where ξ_i is a measure (not necessarily a point measure) which stabilizes; this means that ξ_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 ν_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.