## Berry-Esseen Bounds and Cramer-Type Large Deviations for the Volume Distribution of Poisson Cylinder Processes

- A stationary Poisson cylinder process is composed by a stationary Poisson process of k-dimensional affine subspaces in a d-dimensional Euclidean space (1 <= k < d) which are dilated by a family of independent identically distributed random compact cylinder bases taken from the corresponding orthogonal complement. We study the accuracy of normal approximation of the d-volume of the union set of Poisson cylinders that covers cW when the scaling factor c > 0 becomes large. Here W denotes some fixed compact star-shaped window set containing the origin as inner point. We give lower and upper bounds of the variance of the union set in cW which exhibit long-range dependence within the union set of cylinders. Our main results are sharp estimates of the higher-order cumulants of the d-volume of all Poisson cylinders in cW under the assumption that the (d-k)-volume of the typical cylinder base possesses a finite exponential moment. These estimates enable us to apply the celebrated Lemma on largeA stationary Poisson cylinder process is composed by a stationary Poisson process of k-dimensional affine subspaces in a d-dimensional Euclidean space (1 <= k < d) which are dilated by a family of independent identically distributed random compact cylinder bases taken from the corresponding orthogonal complement. We study the accuracy of normal approximation of the d-volume of the union set of Poisson cylinders that covers cW when the scaling factor c > 0 becomes large. Here W denotes some fixed compact star-shaped window set containing the origin as inner point. We give lower and upper bounds of the variance of the union set in cW which exhibit long-range dependence within the union set of cylinders. Our main results are sharp estimates of the higher-order cumulants of the d-volume of all Poisson cylinders in cW under the assumption that the (d-k)-volume of the typical cylinder base possesses a finite exponential moment. These estimates enable us to apply the celebrated Lemma on large deviations' due to V. Statulevicius.…

Author: | Lothar HeinrichGND, Malte Spiess |
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URN: | urn:nbn:de:bvb:384-opus4-11043 |

Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/1311 |

Series (Serial Number): | Preprints des Instituts für Mathematik der Universität Augsburg (2009-26) |

Type: | Preprint |

Language: | English |

Publishing Institution: | Universität Augsburg |

Contributing Corporation: | Universität Ulm, Institut für Stochastik |

Release Date: | 2009/10/16 |

Tag: | stationary random sets; stochastic geometry; central limit theorem; higher-order (mixed) cumulant; moment- and cumulant generating function |

GND-Keyword: | Stochastische Geometrie; Poisson-Prozess; Zentraler Grenzwertsatz; Große Abweichung; Asymptotische Approximation |

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 |

Licence (German): | Deutsches Urheberrecht mit Print on Demand |