Poisson point process

In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution. The process and the distribution are named after French mathematician Siméon Denis Poisson. The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science. This point process is used as a mathematical model for seemingly random processes in numerous disciplines including astronomy, biology, ecology, geology, seismology, physics, economics, image processing, and telecommunications. The Poisson point process is often defined on the real number line, where it can be viewed as a stochastic process. It is used, for example, in queueing theory to model random events distributed in time, such as the arrival of customers at a store, phone calls at an telephone exchange, or the occurrence of earthquakes. In the plane, the point process—also known as a spatial Poisson process—can represent the locations of scattered objects such as transmitters in a wireless network, particles colliding into a particle detector, or trees in a forest. The process is widely used in mathematical models and in related fields, including spatial point processes, stochastic geometry, spatial statistics and continuum percolation theory. The point process depends on a single mathematical object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings, a Radon measure. In the first case, the constant, known as the rate or intensity, is the average density of the points in the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process. In the second case, the point process is called an inhomogeneous or nonhomogeneous Poisson point process, and the average density of points depend on the location of the underlying space of the Poisson point process. The word point is often omitted, but there are other Poisson processes of objects, which, instead of points, consist of more complicated mathematical objects such as lines and polygons, and such processes can be based on the Poisson point process. Both the homogeneous and nonhomogeneous Poisson point processes are particular cases of the generalized renewal process.

Overview of definitions Depending on the setting, the process has several equivalent definitions as well as definitions of varying generality owing to its many applications and characterizations. The Poisson point process can be defined, studied and used in one dimension, for example, on the real line, where it can be interpreted as a counting process or part of a queueing model; in higher dimensions such as the plane where it plays a role in stochastic geometry and spatial statistics; or on more general mathematical spaces. Consequently, the notation, terminology and level of mathematical rigour used to define and study the Poisson point process and points processes in general vary according to the context. Despite all this, the Poisson point process has two key properties—the Poisson property and the independence property— that play an essential role in all settings where the Poisson point process is used. The two properties are not logically independent; indeed, the Poisson distribution of point counts implies the independence property, while in the converse direction the assumptions that: (i) the point process is simple, (ii) has no fixed atoms, and (iii) is a.s. boundedly finite are required.

Poisson distribution of point counts A Poisson point process is characterized via the Poisson distribution. The Poisson distribution is the probability distribution of a random variable N {\textstyle N} (called a Poisson random variable) such that the probability that N {\displaystyle \textstyle N} equals n {\displaystyle \textstyle n} is given by:

Pr { N = n } = Λ n n ! e − Λ {\displaystyle \Pr\{N=n\}={\frac {\Lambda ^{n}}{n!}}e^{-\Lambda }}

where n ! {\textstyle n!} denotes factorial and the parameter Λ {\textstyle \Lambda } determines the shape of the distribution. (In fact, Λ {\textstyle \Lambda } equals the expected value of N {\textstyle N} .) By definition, a Poisson point process has the property that the number of points in a bounded region of the process's underlying space is a Poisson-distributed random variable.

Complete independence Consider a collection of disjoint and bounded subregions of the underlying space. By definition, the number of points of a Poisson point process in each bounded subregion will be completely independent of all the others. This property is known under several names such as complete randomness, complete independence, or independent scattering and is common to all Poisson point processes. In other words, there is a lack of interaction between different regions and the points in general, which motivates the Poisson process being sometimes called a purely or completely random process.

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