Perfect and Imperfect Simulations in Stochastic Geometry

This thesis presents new developments and applications of simulation methods in stochastic geometry. Simulation is a useful tool for the statistical analysis of spatial point patterns. We use simulation to investigate the power of tests based on the J-function, a new measure of spatial interaction in point patterns. The power of tests based on J is compared to the power of tests based on alternative measures of spatial interaction. Many models in stochastic geometry can only be sampled using Markov chain Monte Carlo methods. We present and extend a new generation of Markov chain Monte Carlo methods, the perfect simulation algorithms. In contrast to conventional Markov chain Monte Carlo methods perfect simulation methods are able to check whether the sampled Markov chain has reached equilibrium yet, thus ensuring that the exact equilibrium distribution is sampled. There are two types of perfect simulation algorithms. Coupling from the Past and Fill’s interruptible algorithm. We present Coupling from the Past in the most general form available and provide a classification of Coupling from the Past algorithms. Coupling from the Past is then extended to produce exact samples of a Boolean model which is conditioned to cover a set of locations with grains. Finally we discuss Fill’s interruptible algorithm and show how to extend the original algorithm to continuous distributions by applying it to a point process example.