On the Variance of the Area of Planar Cylinder Processes Driven by Brillinger-Mixing Point Processes

Abstract: We study some asymptotic properties of cylinder processes in the plane defined as union sets of dilated straight lines (appearing as mutually overlapping infinitely long strips) derived from a stationary independently marked point process on the real line, where the marks describe thickness and orientation of individual cylinders. Such cylinder processes form an important class of (in general non-stationary) planar random sets. We observe the cylinder process in an unboundedly growing domain $\rho K$ when $\rho \to \infty\,$, where the set $K$ is compact and star-shaped w.r.t. the origin ${\bf o}$ being an inner point of $K$. Provided the unmarked point process satisfies a Brillinger-type mixing condition and the thickness of the typical cylinder has a finite second moment we prove a (weak) law of large numbers as well as a formula of the asymptotic variance for the area of the cylinder process in $\rho K$. Due to the long-range dependencies of the cylinder process, this variance increases proportionally to $\rho^3$.