2010/01/06 by Tomasz Schreiber, Schreiber, Tomasz, Christoph Thaele +1 · 2 citations
Mathematics · #52A22 #60D05 #60F05 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #Point processes and geometric inequalities #Probability (math.PR) #math.PR #msc:52A22 #msc:60D05 #msc:60F05
paper · pdf · doi:10.48550/arxiv.1001.0990
51 pages
openalex publication_date 2010/01/06 · arxiv created 2010/03/29 · arxiv updated 2015/03/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 ℝd 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.