An Introduction to the Theory of Point Processes - Probability and Its Applications

Abstract

This last chapter provides an introduction to spatial point processes, meaning for the most part results for point processes in R 2 and R 3 where the order properties of the real line, which governed the development in the preceding chapter, are no longer available. The material we present falls into two main components. In the first four sections we review mainly descriptive properties, distinguishing between distance and directional properties of spatial point patterns, starting from finite models, moving on to the moment properties of line processes, and then revisiting space-time models, where time reappears so that many of the modelling concepts in Chapter 14 are again available, but spatial patterns also play an important role. The three final sections of the chapter provide an introduction to modelling centred around the concept of the Papangelou intensity; we provide some background and motivation from the statistical and physical settings, then attempt an introduction to the more mathematical theory.