Central limit theorems for Poisson hyperplane tessellations

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dc.creatorHeinrich, Lothar
dc.creatorSchmidt, Hendrik
dc.creatorSchmidt, Volker
dc.date2006-07-05
dc.date.accessioned2026-07-07T07:18:03Z
dc.date.available2026-07-07T07:18:03Z
dc.descriptionWe derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in $\mathbb{R}^d$. This result generalizes an earlier one proved by Paroux [Adv. in Appl. Probab. 30 (1998) 640--656] for intersection points of motion-invariant Poisson line processes in $\mathbb{R}^2$. Our proof is based on Hoeffding's decomposition of $U$-statistics which seems to be more efficient and adequate to tackle the higher-dimensional case than the ``method of moments'' used in [Adv. in Appl. Probab. 30 (1998) 640--656] to treat the case $d=2$. Moreover, we extend our central limit theorem in several directions. First we consider $k$-flat processes induced by Poisson hyperplane processes in $\mathbb{R}^d$ for $0\le k\le d-1$. Second we derive (asymptotic) confidence intervals for the intensities of these $k$-flat processes and, third, we prove multivariate central limit theorems for the $d$-dimensional joint vectors of numbers of $k$-flats and their $k$-volumes, respectively, in an increasing spherical region.
dc.descriptionPublished at http://dx.doi.org/10.1214/105051606000000033 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/math/0607120
dc.identifierhttp://arxiv.org/abs/math/0607120
dc.identifierAnnals of Applied Probability 2006, Vol. 16, No. 2, 919-950
dc.identifierdoi:10.1214/105051606000000033
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/114161
dc.subjectProbability
dc.subject60D05 (Primary) 60F05, 62F12 (Secondary)
dc.titleCentral limit theorems for Poisson hyperplane tessellations
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