SECOND-ORDER THEORY FOR ITERATION STABLE TESSELLATIONS

Tomasz Schreiber

Christoph Thäle

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 family of space-time models considered in stochastic geometry. The
previously developed martingale tools are used to study second-order properties of STIT
tessellations. A general formula for the variance of the total surface area of cell boundaries
inside an observation window is shown. This general expression is combined with tools from
integral geometry to derive exact and asymptotic second-order formulas in the stationary and
isotropic regime. Also a general formula for the pair-correlation function of the surface
measure is found.

2000 AMS Mathematics Subject Classification: Primary: 60D05; Secondary: 52A22,
60G55.

Keywords and phrases: Chord-power integral, integral geometry, iteration/nesting,
martingale theory, pair-correlation function, random geometry, random tessellation,
stochastic stability, stochastic geometry.