Moment measure

For some integer ${\displaystyle \textstyle n=1,2,\dots }$, the ${\displaystyle \textstyle n}$-th power of a point process ${\displaystyle \textstyle {N}}$ is defined as:[2]

${\displaystyle {N}^{n}(B_{1}\times \cdots \times B_{n})=\prod _{i=1}^{n}{N}(B_{i})}$

where ${\displaystyle \textstyle B_{1},...,B_{n}}$ is a collection of not necessarily disjoint Borel sets (in ${\displaystyle \textstyle {\textbf {R}}^{d}}$), which form a ${\displaystyle \textstyle n}$-fold Cartesian product of sets denoted by ${\displaystyle B_{1}\times \cdots \times B_{n}}$. The symbol ${\displaystyle \textstyle \Pi }$ denotes standard multiplication.

The notation ${\displaystyle \textstyle {N}(B_{i})}$ reflects the interpretation of the point process ${\displaystyle \textstyle {N}}$ as a random measure.[3]

The ${\displaystyle \textstyle n}$-th power of a point process ${\displaystyle \textstyle {N}}$ can be equivalently defined as:[3]

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

where summation is performed over all ${\displaystyle \textstyle n}$-tuples of (possibly repeating) points, and ${\displaystyle \textstyle \mathbf {1} }$ denotes an indicator function such that ${\displaystyle \textstyle \mathbf {1} _{B_{1}}}$is a Dirac measure. This definition can be contrasted with the definition of the n-factorial power of a point process for which each n-tuples consists of n points.

The ${\displaystyle \textstyle n}$-th moment measure is defined as:

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

where the E denotes the expectation (operator) of the point process ${\displaystyle \textstyle {N}}$. In other words, the n-th moment measure is the expectation of the n-th power of some point process.

The ${\displaystyle \textstyle n\,}$th moment measure of a point process ${\displaystyle \textstyle {N}}$ is equivalently defined[3] as:

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

where ${\displaystyle \textstyle f}$ is any non-negative measurable function on ${\displaystyle \textstyle {\textbf {R}}^{nd}}$ and the sum is over ${\displaystyle \textstyle n}$-tuples of points for which repetition is allowed.

For some Borel set B, the first moment of a point process N is:

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

where ${\displaystyle \textstyle M^{1}}$ is known, among other terms, as the intensity measure[3] or mean measure,[8] and is interpreted as the expected or average number of points of ${\displaystyle \textstyle {N}}$ found or located in the set ${\displaystyle \textstyle B}$.

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

${\displaystyle M^{2}(A\times B)=E[{N}(A){N}(B)],}$

which for a single Borel set ${\displaystyle \textstyle B}$ becomes

${\displaystyle M^{2}(B\times B)=(E[{N}(B)])^{2}+{\text{Var}}[{N}(B)],}$

where ${\displaystyle \textstyle {\text{Var}}[{N}(B)]}$ denotes the variance of the random variable ${\displaystyle \textstyle {N}(B)}$.

The previous variance term alludes to how moments measures, like moments of random variables, can be used to calculate quantities like the variance of point processes. A further example is the covariance of a point process ${\displaystyle \textstyle {N}}$ for two Borel sets ${\displaystyle \textstyle A}$ and ${\displaystyle \textstyle B}$, which is given by:[2]

${\displaystyle {\text{Cov}}[{N}(A),{N}(B)]=M^{2}(A\times B)-M^{1}(A)M^{1}(B)}$