The Poisson Process, Compound Poisson Process, and Poisson Random Field

Poisson processes broadly refer to stochastic processes that are the result of counting occurrences of some random phenomena (points) in time or space such that occurrences of points in disjoint regions are statistically independent, and counts of two or more occurrences in an infinitesimally small region are negligible. This chapter provides the definition and some characteristic properties of both homogeneous and inhomogeneous Poisson processes, and more general random fields; the latter refers to occurrences in non-linearly ordered (e.g., non-temporal) spaces. The compound Poisson process is a fundamentally important example from the perspective of both applications and general representations of processes with independent increments. As such it may be viewed as a continuous parameter generalization of the random walk.