Point process notation

The notation, as well as the terminology, of point processes depends on their setting and interpretation as mathematical objects which under certain assumptions can be interpreted as random sequences of points, random sets of points or random counting measures.[1]

The different interpretations of point processes as random sets and counting measures is captured with the often used notation [1][3][8][11] in which:

• ${\displaystyle {N}}$ denotes a set of random points.
• ${\displaystyle {N}(B)}$ denotes a random variable that gives the number of points of ${\displaystyle {N}}$ in ${\displaystyle B}$ (hence it is a random counting measure).

Denoting the counting measure again with ${\displaystyle \#}$, this dual notation implies:

${\displaystyle {N}(B)=\#(B\cap {N}).}$

If ${\displaystyle f}$ is some measurable function on R^d, then the sum of ${\displaystyle f(x)}$ over all the points ${\displaystyle x}$ in ${\displaystyle {N}}$ can be written in a number of ways [1][3] such as:

${\displaystyle f(x_{1})+f(x_{2})+\cdots }$

which has the random sequence appearance, or with set notation as:

${\displaystyle \sum _{x\in {N}}f(x)}$

or, equivalently, with integration notation as:

${\displaystyle \int _{{\textbf {R}}^{d}}f(x){N}(dx)}$

where which puts an emphasis on the interpretation of ${\displaystyle {N}}$ being a random counting measure. An alternative integration notation may be used to write this integral as:

${\displaystyle \int _{{\textbf {R}}^{d}}f\,d{N}}$

The dual interpretation of point processes is illustrated when writing the number of ${\displaystyle {N}}$ points in a set ${\displaystyle B}$ as:

${\displaystyle {N}(B)=\sum _{x\in {N}}1_{B}(x)}$

where the indicator function ${\displaystyle 1_{B}(x)=1}$ if the point ${\displaystyle x}$ is exists in ${\displaystyle B}$ and zero otherwise, which in this setting is also known as a Dirac measure.[11] In this expression the random measure interpretation is on the left-hand side while the random set notation is used is on the right-hand side.

The average or expected value of a sum of functions over a point process is written as:[1][3]

${\displaystyle E\left[\sum _{x\in {N}}f(x)\right]\qquad {\text{or}}\qquad \int _{\textbf {N}}\sum _{x\in {N}}f(x)P(d{N}),}$

where (in the random measure sense) ${\displaystyle P}$ is an appropriate probability measure defined on the space of counting measures ${\displaystyle {\textbf {N}}}$. The expected value of ${\displaystyle {N}(B)}$ can be written as:[1]

${\displaystyle E[{N}(B)]=E\left(\sum _{x\in {N}}1_{B}(x)\right)\qquad {\text{or}}\qquad \int _{\textbf {N}}\sum _{x\in {N}}1_{B}(x)P(d{N}).}$

which is also known as the first moment measure of ${\displaystyle {N}}$. The expectation of such a random sum, known as a shot noise process in the theory of point processes, can be calculated with Campbell's theorem.[2]

Point processes are employed in other mathematical and statistical disciplines, hence the notation may be used in fields such stochastic geometry, spatial statistics or continuum percolation theory, and areas which use the methods and theory from these fields.

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