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

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 rho K as rho -> infinity. In the case the unmarked point process is Brillinger mixing, we prove themean-square convergence of the area fraction of the cylinder process in rho K. Under stronger versions of Brillinger mixing, we obtain the exact variance asymptotics of the area of the cylinder process in rho K as rho -> infinity. Due to the long-range dependence of the cylinder process, this variance increases asymptotically proportionally to rho(3).