## Central Limit Theorems for Motion-Invariant Poisson Hyperplanes in Expanding Convex Bodies

- We consider motion-invariant (i.e. stationary and isotropic) Poisson hyperplane processes subdividing the d-dimensional Euclidean space into a collection of convex d-polytopes. Among others we prove that the total number of vertices of these polytopes lying in an unboundedly growing convex body (with inner points) is asymptotically normally distributed. We are able to extend this central limit theorem to the total s-volume of the s-flats of all these polytopes contained in the expanding convex window, where s runs between 1 and d-1. It is noteworthy that the variances of these statistics are shown to be asymptotically proportional to some motion-invariant, non-decreasing, non-additive ovoid functional of the convex sampling body. Estimates and extremal properties of these functionals are of interest in convex geometry. Due to long-range dependences between distant parts of the generated Poisson hyperplane tessellation the variances increase faster than the volume of the samplingWe consider motion-invariant (i.e. stationary and isotropic) Poisson hyperplane processes subdividing the d-dimensional Euclidean space into a collection of convex d-polytopes. Among others we prove that the total number of vertices of these polytopes lying in an unboundedly growing convex body (with inner points) is asymptotically normally distributed. We are able to extend this central limit theorem to the total s-volume of the s-flats of all these polytopes contained in the expanding convex window, where s runs between 1 and d-1. It is noteworthy that the variances of these statistics are shown to be asymptotically proportional to some motion-invariant, non-decreasing, non-additive ovoid functional of the convex sampling body. Estimates and extremal properties of these functionals are of interest in convex geometry. Due to long-range dependences between distant parts of the generated Poisson hyperplane tessellation the variances increase faster than the volume of the sampling window. The proving technique used is based on Hoeffding's decomposition of U-statistics with Poisson distributed sample size. The obtained results generalize earlier ones proved in the special case of growing spherical windows.…

Author: | Lothar HeinrichGND |
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URN: | urn:nbn:de:bvb:384-opus4-4851 |

Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/609 |

Series (Serial Number): | Preprints des Instituts für Mathematik der Universität Augsburg (2007-43) |

Type: | Preprint |

Language: | English |

Publishing Institution: | Universität Augsburg |

Release Date: | 2007/12/27 |

Tag: | Poisson hyperplane tessellation; Crofton's formula; U-statistics; nonadditive ovoid functionals |

GND-Keyword: | U-Statistik; Poisson-Hyperebenenmosaik; Ovoid |

Institutes: | Mathematisch-Naturwissenschaftlich-Technische Fakultät |

Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik | |

Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Stochastik und ihre Anwendungen | |

Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |