Models Constructed via Conditioning: Cox, Cluster, and Marked Point Processes

In this chapter, we bring together a number of the most widely used classes of point process models. Their common theme is the generation of the final model by a two-stage construction: first, the generation of an indexed family of processes, and then an operation applied to members of the family to produce the final process. The first two classes (Cox and cluster processes) extend the simple Poisson process in much the same way that the mixed and compound Poisson distributions extend the basic Poisson distribution. Independence plays a central role and leads to elegant results for moment and generating functional relationships. Both processes are used typically in contexts where the realizations are stationary and therefore define infinite collections of points. To deal with these issues, we anticipate the transition from finite to general point processes to be carried out in Chapter 9 and present in Section 6.1 a short review of some key results for more general point processes and random measures. The third class of processes considered in this chapter represents a generalization in a different direction. In many situations, events are characterized by both a location and a weight or other distinguishing attribute. Such processes are already covered formally by the general theory, as they can be represented as a special type of point process on a product space. However, marked point processes are deserving of study in their own right because of their wide range of applications, such as in queueing theory, and their conceptual importance in contexts such as Palm theory (see [MKM] especially).