Advances in Applied Probability

Central limit theorems for volume and surface content of stationary Poisson cylinder processes in expanding domains

Lothar Heinrich and Malte Spiess

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Abstract

A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed of a stationary Poisson process of k-flats (0 ≤ k ≤ d-1) which are dilated by independent and identically distributed random compact cylinder bases taken from the corresponding (d-k)-dimensional orthogonal complement. If the second moment of the (d-k)-volume of the typical cylinder base exists, we prove asymptotic normality of the d-volume of the union set of Poisson cylinders that covers an expanding star-shaped domain ϱ W as ϱ grows unboundedly. Due to the long-range dependencies within the union set of cylinders, the variance of its d-volume in ϱ W increases asymptotically proportional to the (d+k)th power of ϱ. To obtain the exact asymptotic behaviour of this variance, we need a distinction between discrete and continuous directional distributions of the typical k-flat. A corresponding central limit theorem for the surface content is stated at the end.

Article information

Source
Adv. in Appl. Probab., Volume 45, Number 2 (2013), 312-331.

Dates
First available in Project Euclid: 10 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.aap/1370870120

Digital Object Identifier
doi:10.1239/aap/1370870120

Mathematical Reviews number (MathSciNet)
MR3102453

Zentralblatt MATH identifier
1282.60012

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems
Secondary: 60F10: Large deviations 60G55: Point processes

Keywords
Independently marked Poisson process truncated typical cylinder direction space volume fraction moment convergence theorem asymptotic variance long-range dependence higher-order (mixed) cumulant

Citation

Heinrich, Lothar; Spiess, Malte. Central limit theorems for volume and surface content of stationary Poisson cylinder processes in expanding domains. Adv. in Appl. Probab. 45 (2013), no. 2, 312--331. doi:10.1239/aap/1370870120. https://projecteuclid.org/euclid.aap/1370870120


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