In statistics and probability theory, a point process or point field is a set of a random number of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Point processes on the real line form an important special case that is particularly amenable to study, because the points are ordered in a natural way, and the whole point process can be described completely by the (random) intervals between the points. These point processes are frequently used as models for random events in time, such as the arrival of customers in a queue (queueing theory), of impulses in a neuron (computational neuroscience), particles in a Geiger counter, location of radio stations in a telecommunication network or of searches on the world-wide web. General point processes on a Euclidean space can be used for spatial data analysis, which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience, economics and others.
Conventions Since point processes were historically developed by different communities, there are different mathematical interpretations of a point process, such as a random counting measure or a random set, and different notations. The notations are described in detail on the point process notation page. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. Others consider a point process as a stochastic process, where the process is indexed by sets of the underlying space on which it is defined, such as the real line or n {\displaystyle n} -dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes. Sometimes the term "point process" is not preferred, as historically the word "process" denoted an evolution of some system in time, so point process is also called a random point field.
Mathematics In mathematics, a point process is a random element whose values are "point patterns" on a set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting measure, it is sufficient for more applied purposes to think of a point pattern as a countable subset of S that has no limit points.
Definition To define general point processes, we start with a probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} , and a measurable space ( S , S ) {\displaystyle (S,{\mathcal {S}})} where S {\displaystyle S} is a locally compact second countable Hausdorff space and S {\displaystyle {\mathcal {S}}} is its Borel σ-algebra. Consider now an integer-valued locally finite kernel ξ {\displaystyle \xi }
from ( Ω , F ) {\displaystyle (\Omega ,{\mathcal {F}})} into ( S , S ) {\displaystyle (S,{\mathcal {S}})} , that is, a mapping
Ω × S ↦ Z + {\displaystyle \Omega \times {\mathcal {S}}\mapsto \mathbb {Z} _{+}} such that:
For every ω ∈ Ω {\displaystyle \omega \in \Omega } , ξ ( ω , ⋅ ) {\displaystyle \xi (\omega ,\cdot )} is a (integer-valued) locally finite measure on S {\displaystyle S} . For every B ∈ S {\displaystyle B\in {\mathcal {S}}} , ξ ( ⋅ , B ) : Ω → Z + {\displaystyle \xi (\cdot ,B):\Omega \to \mathbb {Z} _{+}} is a random variable over Z + {\displaystyle \mathbb {Z} _{+}} . This kernel defines a random measure in the following way. We would like to think of ξ {\displaystyle \xi }
as defining a mapping which maps ω ∈ Ω {\displaystyle \omega \in \Omega } to a measure ξ ω ∈ M ( S ) {\displaystyle \xi _{\omega }\in {\mathcal {M}}({\mathcal {S}})}
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