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.

Published 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 (2)

• Gaussian limits for random measures in geometric probability
• Explicit laws of large numbers for random nearest-neighbour type graphs

Cited by in corpus (7)

• Nonparametric estimation of surface integrals
• Gaussian limits for generalized spacings
• Poisson--Voronoi approximation
• Moderate deviations for some point measures in geometric probability
• Stabilization and limit theorems for geometric functionals of Gibbs point processes
• Limit theory for planar Gilbert tessellations
• Normal approximation of Gibbsian sums in geometric probability