Second-Order Theory for Iteration Stable Tessellations

Abstract

This paper deals with iteration stable (STIT) tessellations, and, more generally, with a certain class of tessellations that are infinitely divisible with respect to iteration. They form a new, rich and flexible class of spatio-temporal models considered in stochastic geometry. The martingale tools developed in \cite{STP1} are used to study second-order properties of STIT tessellations. Firstly, a general formula for the variance of the total surface area of cell boundaries inside a convex observation window is shown. This general expression is combined with tools from integral geometry to derive explicit exact and asymptotic second-order formulae in the stationary and isotropic set-up, where a family of chord-power integrals plays an important role. Also a general formula for the pair-correlation function of the surface measure is found.