In this thesis, we consider stationary random mosaics with a focus on the Poisson-Voronoi mosaic and the Poisson-Delaunay mosaic. We consider properties of stationary random mosaics in R2, such as mean value results of the typical cell. Further, we simulate various mean value results of the typical cell, a random neighbor of the typical cell, and the zero cell for the Poisson-Voronoi mosaic in R2. Some theory of point processes is introduced that is needed for random mosaics, including Palm theory, marked point processes, and the Pois point process. Finally, we consider an incremental flip-based algorithm for generating the Voronoi mosaic.

Statistical aspects of random mosaics have not been heretofore given enough attention. This thesis deals with the derivation of estimators and statistical tests in a three-dimensional Poisson-Voronoi mosaic model. The first chapter compiles elementary results in the fields of point processes, random closed sets and particle processes. These are used in a second chapter to deduce geometric properties of random mosaics. The third chapter introduces the statistical research itself, estimators and model tests. Horvitz- Thompson estimator is introduced in order to correct statistics calculated on a reduced sample. Own results are tried in a computer simulation and compared to existing research in the last chapter. Mainly, the quality of estimators and the power of proposed tests is observed. 1