  Since the seminal work of Mecke, Nagel and Weiss, the iteration stable (STIT)
tessellations have attracted considerable interest in stochastic geometry as a
natural and flexible yet analytically tractable model for hierarchical spatial
cell-splitting and crack-formation processes. The purpose of this paper is to
describe large scale asymptotic geometry of STIT tessellations in
$\mathbb{R}^d$ and more generally that of non-stationary iteration infinitely
divisible tessellations. We study several aspects of the typical first-order
geometry of such tessellations resorting to martingale techniques as providing
a direct link between the typical characteristics of STIT tessellations and
those of suitable mixtures of Poisson hyperplane tessellations. Further, we
also consider second-order properties of STIT and iteration infinitely
divisible tessellations, such as the variance of the total surface area of cell
boundaries inside a convex observation window. Our techniques, relying on
martingale theory and tools from integral geometry, allow us to give explicit
and asymptotic formulae. Based on these results, we establish a functional
central limit theorem for the length/surface increment processes induced by
STIT tessellations. We conclude a central limit theorem for total edge
length/facet surface, with normal limit distribution in the planar case and
non-normal ones in all higher dimensions.
