• Bernoulli
  • Volume 13, Number 4 (2007), 1124-1150.

Laws of large numbers in stochastic geometry with statistical applications

Mathew D. Penrose

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Given n independent random marked d-vectors (points) Xi distributed with a common density, define the measure νn=∑iξi, where ξi is a measure (not necessarily a point measure) which stabilizes; this means that ξi is determined by the (suitably rescaled) set of points near Xi. For bounded test functions f on Rd, we give weak and strong laws of large numbers for νn(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 Xi 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.

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Bernoulli, Volume 13, Number 4 (2007), 1124-1150.

First available in Project Euclid: 9 November 2007

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law of large numbers nearest neighbours nonparametric regression point process random measure stabilization Voronoi coverage


Penrose, Mathew D. Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 (2007), no. 4, 1124--1150. doi:10.3150/07-BEJ5167.

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