Bernoulli

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

Laws of large numbers in stochastic geometry with statistical applications

Mathew D. Penrose

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 9 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1194625605

Digital Object Identifier
doi:10.3150/07-BEJ5167

Mathematical Reviews number (MathSciNet)
MR2364229

Zentralblatt MATH identifier
1143.60013

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

Citation

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. https://projecteuclid.org/euclid.bj/1194625605


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