Chapter 1 Point processes

Abstract

A point process is a model for describing the random numbers of occurrences of a certain event in time intervals or of the numbers of points in regions of a space. This chapter describes the structure of point processes and discusses their basic properties. The major families of point processes are (1) Poisson, compound Poisson, and Cox processes, (2) infinitely divisible and independent increment point processes, (3) renewal processes and processes defined by interval properties, (4) stationary point processes, (5) sample processes and cluster processes, (5) point processes related to martingale theory and stochastic calculus. Marked point processes are vehicles for modeling compound point processes where points occur in batches and for modeling random parameters or operations associated with the points of a process. An important class of marks are such that a mark for a point is a random variable (or element) possibly depending on the location of the point, but is independent of everything else. The focus of the chapter is on presenting tools for modeling stochastic systems rather than on applications of the tools.