  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 $\rho^3$.
