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 ${\nu }_n={\sum }_i{\xi }_i$, where ${\xi }_i$ is a measure (not necessarily a point measure) which stabilizes; this means that ${\xi }_i$ 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 ${\nu }_n(f)$. 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.

arXiv commentPublished in at http://dx.doi.org/10.3150/07-BEJ5167 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Gaussian limits for random measures in geometric probability
• On the total length of the random minimal directed spanning tree
• Explicit laws of large numbers for random nearest-neighbour type graphs

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• Poisson--Voronoi approximation
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• Multivariate normal approximation in geometric probability
• Asymptotic theory for statistics of the Poisson--Voronoi approximation
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• Limit theorems for random spatial drainage networks
• Typical Geometry, Second-Order Properties and Central Limit Theory for Iteration Stable Tessellations
• Solving Stochastic Inverse Problems using Sigma-Algebras on Contour Maps
• Stabilization and limit theorems for geometric functionals of Gibbs point processes
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• Limit theory for planar Gilbert tessellations
• Asymptotic results for stabilizing functionals of point processes having fast decay of correlations
• A remark on the convergence of Betti numbers in the thermodynamic regime
• Normal approximation of Gibbsian sums in geometric probability