Variance asymptotics for the area of planar cylinder processes generated by Brillinger-mixing point processes

  • We introduce cylinder processes in the plane defined as union sets of dilated straight lines (appearing as mutually overlapping infinitely long strips) generated by a stationary independently marked point process on the real line, where the marks describe the width and orientation of the individual cylinders. We study the behavior of the total area of the union of strips contained in a space-filling window ϱK as ϱ → ∞. In the case the unmarked point process is Brillinger mixing, we prove themean-square convergence of the area fraction of the cylinder process in ϱK. Under stronger versions of Brillinger mixing, we obtain the exact variance asymptotics of the area of the cylinder process in ϱK as ϱ → ∞. Due to the long-range dependence of the cylinder process, this variance increases asymptotically proportionally to ϱ3.

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Author:Daniela Flimmel, Lothar HeinrichGND
Frontdoor URL
Parent Title (English):Lithuanian Mathematical Journal
Publisher:Springer Science and Business Media LLC
Place of publication:Berlin
Year of first Publication:2023
Publishing Institution:Universität Augsburg
Release Date:2023/05/02
Tag:General Mathematics
First Page:58
Last Page:80
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):CC-BY 4.0: Creative Commons: Namensnennung (mit Print on Demand)