n th factorial power of a point process

For some positive integer ${\displaystyle \textstyle n=1,2,\ldots }$, the ${\displaystyle \textstyle n}$-th factorial power of a point process ${\displaystyle \textstyle {N}}$ on ${\displaystyle \textstyle {\textbf {R}}^{d}}$ is defined as:^[2]

${\displaystyle {N}^{(n)}(B_{1}\times \cdots \times B_{n})=\sum _{(x_{1}\neq \dots \neq x_{n})\in {N}}\prod _{i=1}^{n}\mathbf {1} _{B_{i}}(x_{i})}$

where ${\displaystyle \textstyle B_{1},...,B_{n}}$ is a collection of not necessarily disjoint Borel sets in ${\displaystyle \textstyle {\textbf {R}}^{d}}$, which form an ${\displaystyle \textstyle n}$-fold Cartesian product of sets denoted by:

${\displaystyle B_{1}\times \cdots \times B_{n}.}$

The symbol ${\displaystyle \textstyle \mathbf {1} }$ denotes an indicator function such that ${\displaystyle \textstyle \mathbf {1} _{B_{1}}}$is a Dirac measure for the set ${\displaystyle \textstyle B_{n}}$. The summation in the above expression is performed over all ${\displaystyle \textstyle n}$-tuples of distinct points, including permutations, which can be contrasted with the definition of the n-th power of a point process. The symbol ${\displaystyle \textstyle \Pi }$ denotes multiplication while the existence of various point process notation means that the n-th factorial power of a point process is sometimes defined using other notation.^[2]

n th factorial moment measure

The n th factorial moment measure or n th order factorial moment measure is defined as:

${\displaystyle M^{(n)}(B_{1}\times \cdots \times B_{n})=E[{N}^{(n)}(B_{1}\times \cdots \times B_{n})],}$

where the E denotes the expectation (operator) of the point process N. In other words, the n-th factorial moment measure is the expectation of the n th factorial power of some point process.

The n th factorial moment measure of a point process N is equivalently defined^[3] by:

${\displaystyle \int _{{\textbf {R}}^{nd}}f(x_{1},\ldots ,x_{n})M^{(n)}(dx_{1},\ldots ,dx_{n})=E\left[\sum _{(x_{1}\neq \cdots \neq x_{n})\in {N}}f(x_{1},\ldots ,x_{n})\right],}$

where ${\displaystyle f}$ is any non-negative measurable function on ${\displaystyle {\textbf {R}}^{nd}}$, and the above summation is performed over all ${\displaystyle n}$ tuples of distinct points, including permutations. Consequently, the factorial moment measure is defined such that there are no points repeating in the product set, as opposed to the moment measure.^[7]

First factorial moment measure

The first factorial moment measure ${\displaystyle \textstyle M^{1}}$ coincides with the first moment measure:^[2]

${\displaystyle M^{(1)}(B)=M^{1}(B)=E[{N}(B)],}$

where ${\displaystyle \textstyle M^{1}}$ is known, among other terms, as the intensity measure^[3] or mean measure,^[10] and is interpreted as the expected number of points of ${\displaystyle \textstyle {N}}$ found or located in the set ${\displaystyle \textstyle B}$

Second factorial moment measure

The second factorial moment measure for two Borel sets ${\displaystyle \textstyle A}$ and ${\displaystyle \textstyle B}$ is:

${\displaystyle M^{(2)}(A\times B)=M^{2}(A\times B)-M^{1}(A\cap B).}$

Poisson point process

For a general Poisson point process with intensity measure ${\displaystyle \textstyle \Lambda }$ the ${\displaystyle \textstyle n}$-th factorial moment measure is given by the expression:^[3]

${\displaystyle M^{(n)}(B_{1}\times \cdots \times B_{n})=\prod _{i=1}^{n}[\Lambda (B_{i})],}$

where ${\displaystyle \textstyle \Lambda }$ is the intensity measure or first moment measure of ${\displaystyle \textstyle {N}}$, which for some Borel set ${\displaystyle \textstyle B}$ is given by:

${\displaystyle \Lambda (B)=M^{1}(B)=E[{N}(B)].}$

For a homogeneous Poisson point process the ${\displaystyle \textstyle n}$-th factorial moment measure is simply:^[2]

${\displaystyle M^{(n)}(B_{1}\times \cdots \times B_{n})=\lambda ^{n}\prod _{i=1}^{n}|B_{i}|,}$

where ${\displaystyle \textstyle |B_{i}|}$ is the length, area, or volume (or more generally, the Lebesgue measure) of ${\displaystyle \textstyle B_{i}}$. Furthermore, the ${\displaystyle \textstyle n}$-th factorial moment density is:^[3]

${\displaystyle \mu ^{(n)}(x_{1},\ldots ,x_{n})=\lambda ^{n}.}$

The pair-correlation function of the homogeneous Poisson point process is simply

${\displaystyle \rho (x_{1},x_{2})=1,}$

which reflects the lack of interaction between points of this point process.