Cluster Processes, Infinitely Divisible Processes, and Doubly Stochastic Processes

In this chapter we discuss two of the most widely used classes of point processes in application and associate with the first of them the important theoretical problem of determining the structure of infinitely divisible point processes. Each class contains many special cases that have been the subject of extensive analyses in their own right. A key feature of both classes is that their most important members are derivatives of the Poisson process. They are, indeed, natural extensions of the compound and mixed Poisson processes of the previous chapter: the Poisson cluster process extends the notion of the compound Poisson process and the doubly stochastic Poisson process extends the notion of the mixed Poisson process. Because of this feature, both can be handled compactly by the p.g.fl. techniques introduced in Chapter 7, and we make extensive use of this approach. It should be borne in mind, however, that the main advantage of this approach lies precisely in its compactness: it quickly summarizes information that can still be derived quite readily without it and that in less tractable examples may not be so easily expressible in p.g.fl. form.