Point Processes

Abstract

Loosely speaking, a point process is a random ‘discrete’ set of points in some Polish space. Thus, one could use a point process to model experiments like throwing grains of sand onto the floor and noting their locations, or pointing an astronomical telescope in a random direction and noting the positions of the stars seen in the field of view. A mathematical example would be the random set of values taken by a finite sequence of random variables. This latter example makes it clear that we may want to generalize the notion of sets to allow a given point to appear more than once. It turns out that there is a nice mathematical way to accommodate the generalization using a certain class of ℤ̄⁺ -valued measures. The relevant definitions and basic facts are given in the first section. The most important point processes are ‘Poisson point processes’, which are characterized by the property that their intersections with disjoint subsets of the underlying Polish space are independent. These are treated in Sections 3 and 4. An important tool for studying the distributions of point processes is introduced in the fourth section. This tool is needed in the final two sections of the chapter, where various operations on point processes are studied. In particular, the convergence in distribution of point processes is considered in the final section. One nice result from that section is that the Poisson point processes arise as limits of certain naturally defined sequences.