Characteristics of Poisson cylinder processes and their estimation
FacultiesFakultät für Mathematik und Wirtschaftswissenschaften
In this thesis, we consider stationary Poisson cylinder processes (PCPs). Here, a cylinder is defined as the Minkowski sum of a linear vectorspace and a polyconvex set in the orthogonal space. A PCP is a random set of cylinders (with Poisson counting measures). After the introduction and some basic notions, we begin with some basic characteristics for PCPs, namely the capacity functional and related functionals and the specific surface area. As the next result, a central limit theorem for the empirical volume fraction is derived with Berry-Esseen bounds and Cramér-type large deviation results using the method of cumulants. Explicit formulae for the asymptotic variance are calculated. Finally, we consider an estimator for the directional distribution of PCPs which is based on counting the intersection points of the PCP with test hyperplanes in a bounded observation window. To derive the directional distribution from this data, it is necessary to solve the inverse problem of inverting the cosine transform, for which we use the method of the approximate inverse to get numerically stable results. For the resulting estimator, strong consistency in the supremum norm and a central limit theorem for the pointwise convergence with Berry-Esseen bounds and large deviation results are proven. We also derive integral formulae for the variance. Various simulation studies are included. Each chapter ends with a short discussion and open problems.
Subject HeadingsPoisson-Prozess [GND]
Stochastic geometry [LCSH]