Moment measure

Definitions

n-th power of a point process

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

${\displaystyle N^n(B_1\times \cdots \times B_n)=\prod _{i=1}^{n}N(B_i)}$

where ${\displaystyle B_1,...,B_n}$ is a collection of not necessarily disjoint Borel sets (in ${\displaystyle {\mathbf{\text{R}}}^d}$), which form a ${\displaystyle n}$-fold Cartesian product of sets denoted by ${\displaystyle B_1\times \cdots \times B_n}$. The symbol ${\displaystyle \mathrm{\Pi }}$ denotes standard multiplication.

The notation ${\displaystyle N(B_i)}$ reflects the interpretation of the point process ${\displaystyle N}$ as a random measure.

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

${\displaystyle N^n(B_1\times \cdots \times B_n)=\sum _{(x_1,\dots ,x_n)\in N}\prod _{i=1}^n{\mathbf{1}}_{B_i}(x_i)}$

where summation is performed over all ${\displaystyle n}$-tuples of (possibly repeating) points, and ${\displaystyle \mathbf{1}}$ denotes an indicator function such that ${\displaystyle {\mathbf{1}}_{B_1}}$ 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 distinct points.

n-th moment measure

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

${\displaystyle M^n(B_1\times \dots \times B_n)=E[N^n(B_1\times \dots \times B_n)],}$

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

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

${\displaystyle {\int }_{{\mathbf{\text{R}}}^{nd}}f(x_1,\dots ,x_n)M^n(d{x}_1,\dots ,d{x}_n)=E\left[\sum _{(x_1,\dots ,x_n)\in N}f(x_1,\dots ,x_n)\right],}$

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

First moment measure

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

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

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

Second moment measure

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

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

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

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

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

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 ${\displaystyle N}$ for two Borel sets ${\displaystyle A}$ and ${\displaystyle B}$, which is given by:

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

Example: Poisson point process

For a general Poisson point process with intensity measure ${\displaystyle \mathrm{\Lambda }}$ the first moment measure is:

${\displaystyle M^1(B)=\mathrm{\Lambda }(B),}$

which for a homogeneous Poisson point process with constant intensity ${\displaystyle \lambda }$ means:

${\displaystyle M^1(B)=\lambda |B|,}$

where ${\displaystyle |B|}$ is the length, area or volume (or more generally, the Lebesgue measure) of ${\displaystyle B}$.

For the Poisson case with measure ${\displaystyle \mathrm{\Lambda }}$ the second moment measure defined on the product set ${\displaystyle (B\times B)}$ is:

${\displaystyle M^2(B\times B)=\mathrm{\Lambda }(B)+\mathrm{\Lambda }(B{)}^2.}$

which in the homogeneous case reduces to

${\displaystyle M^2(B\times B)=\lambda |B|+(\lambda |B|{)}^2.}$