Poisson point process
Notation

The notation of the Poisson point process depends on its setting and the field it is being applied in. For example, on the real line, the Poisson process, both homogeneous or inhomogeneous, is sometimes interpreted as a counting process, and the notation { N (t), t ≥ 0 } {\displaystyle \textstyle \{N(t),t\geq 0\}} is used to represent the Poisson process.

Another reason for varying notation is due to the theory of point processes, which has a couple of mathematical interpretations. For example, a simple Poisson point process may be considered as a random set, which suggests the notation x ∈ N {\displaystyle \textstyle x\in N}, implying that x {\displaystyle \textstyle x} is a random point belonging to or being an element of the Poisson point process N {\displaystyle \textstyle N}. Another, more general, interpretation is to consider a Poisson or any other point process as a random counting measure, so one can write the number of points of a Poisson point process N {\displaystyle \textstyle {N}} being found or located in some (Borel measurable) region B {\displaystyle \textstyle B} as N (B) {\displaystyle \textstyle N(B)}, which is a random variable. These different interpretations results in notation being used from mathematical fields such as measure theory and set theory.

For general point processes, sometimes a subscript on the point symbol, for example x {\displaystyle \textstyle x}, is included so one writes (with set notation) x i ∈ N {\displaystyle \textstyle x_{i}\in N} instead of x ∈ N {\displaystyle \textstyle x\in N}, and x {\displaystyle \textstyle x} can be used for the bound variable in integral expressions such as Campbell's theorem, instead of denoting random points. Sometimes an uppercase letter denotes the point process, while a lowercase denotes a point from the process, so, for example, the point x {\displaystyle \textstyle x} or x i {\displaystyle \textstyle x_{i}} belongs to or is a point of the point process X {\displaystyle \textstyle X}, and be written with set notation as x ∈ X {\displaystyle \textstyle x\in X} or x i ∈ X {\displaystyle \textstyle x_{i}\in X}.

Furthermore, the set theory and integral or measure theory notation can be used interchangeably. For example, for a point process N {\displaystyle \textstyle N} defined on the Euclidean state space R d {\displaystyle \textstyle {\mathbb {R} ^{d}}} and a (measurable) function f {\displaystyle \textstyle f} on R d {\displaystyle \textstyle \mathbb {R} ^{d}}, the expression

demonstrates two different ways to write a summation over a point process (see also Campbell's theorem (probability)). More specifically, the integral notation on the left-hand side is interpreting the point process as a random counting measure while the sum on the right-hand side suggests a random set interpretation.
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