## Bernoulli

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

### Laws of large numbers in stochastic geometry with statistical applications

Mathew D. Penrose

#### 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

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

Digital Object Identifier
doi:10.3150/07-BEJ5167

Mathematical Reviews number (MathSciNet)
MR2364229

Zentralblatt MATH identifier
1143.60013

#### 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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