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

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 $\rho \to \infty,$,wheretheset$K$iscompactandstar-shapedw\mathrm{.}r\mathrm{.}t\mathrm{.}theorigin${\bf o}$beinganinnerpointof$K$. 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 ${\rho }^3$.