Author:
Flimmel Daniela,Heinrich Lothar
Abstract
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.
Publisher
Springer Science and Business Media LLC
Reference21 articles.
1. C. Betken, M. Schulte, and C. Thäle, Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes, 2022, arXiv:2111.04608.
2. D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Vol. I, Springer, New York, 2003.
3. D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Vol. II, Springer, New York, 2008.
4. R.D. Davies, Testing a hypothesis that a point process is Poisson, Adv. Appl. Probab., 9:724–746, 1977.
5. D. Flimmel and L. Heinrich, On the variance of the area of planar cylinder processes driven by Brillinger-mixing point processes, 2021, arXiv:2104.10224.