The Poisson Point Process

Abstract

Properties of multidimensional Poisson point processes (PPPs) are discussed using a constructive approach readily accessible to a broad audience. The processes are defined in terms of a two-step simulation procedure, and their fundamental properties are derived from the simulation. This reverses the traditional exposition, but it enables those new to the subject to understand quickly what PPPs are about, and to see that general nonhomogeneous processes are little more conceptually difficult than homogeneous processes. After reviewing the basic concepts on continuous spaces, several important and useful operations that map PPPs into other PPPs are discussed—these include superposition, thinning, nonlinear transformation, and stochastic transformation. Following these topics is an amusingly provocative demonstration that PPPs are “inevitable.” The chapter closes with a discussion of PPPs whose points lie in discrete spaces and in discrete-continuous spaces. In contrast to PPPs on continuous spaces, realizations of PPPs in these spaces often sample the discrete points repeatedly. This is important in applications such as multitarget tracking.

Make things as simple as possible, but not simpler. ¹

Albert Einstein (paraphrased),

On the Method of Theoretical Physics, 1934

¹What he really said [27]: “It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.”