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February 2014 On the local approximation of mean densities of random closed sets
Elena Villa
Bernoulli 20(1): 1-27 (February 2014). DOI: 10.3150/12-BEJ474


Mean density of lower dimensional random closed sets, as well as the mean boundary density of full dimensional random sets, and their estimation are of great interest in many real applications. Only partial results are available so far in current literature, under the assumption that the random set is either stationary, or it is a Boolean model, or it has convex grains. We consider here non-stationary random closed sets (not necessarily Boolean models), whose grains have to satisfy some general regularity conditions, extending previous results. We address the open problem posed in (Bernoulli 15 (2009) 1222–1242) about the approximation of the mean density of lower dimensional random sets by a pointwise limit, and to the open problem posed by Matheron in (Random Sets and Integral Geometry (1975) Wiley) about the existence (and its value) of the so-called specific area of full dimensional random closed sets. The relationship with the spherical contact distribution function, as well as some examples and applications are also discussed.


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Elena Villa. "On the local approximation of mean densities of random closed sets." Bernoulli 20 (1) 1 - 27, February 2014.


Published: February 2014
First available in Project Euclid: 22 January 2014

zbMATH: 1291.60025
MathSciNet: MR3160572
Digital Object Identifier: 10.3150/12-BEJ474

Rights: Copyright © 2014 Bernoulli Society for Mathematical Statistics and Probability


Vol.20 • No. 1 • February 2014
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