Random measure

In

Random measures can be defined as transition kernels or as random elements. Both definitions are equivalent. For the definitions, let ${\displaystyle E}$ be a separable complete metric space and let ${\displaystyle {\mathcal {E}}}$ be its Borel ${\displaystyle \sigma }$-algebra. (The most common example of a separable complete metric space is ${\displaystyle \mathbb {R} ^{n}}$)

As a transition kernel

A random measure ${\displaystyle \zeta }$ is a (a.s.) locally finite transition kernel from a (abstract) probability space ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ to ${\displaystyle (E,{\mathcal {E}})}$.^[3]

Being a transition kernel means that

• For any fixed ${\displaystyle B\in {\mathcal {\mathcal {E}}}}$, the mapping

${\displaystyle \omega \mapsto \zeta (\omega ,B)}$

is measurable from ${\displaystyle (\Omega ,{\mathcal {A}})}$ to ${\displaystyle (E,{\mathcal {E}})}$

• For every fixed ${\displaystyle \omega \in \Omega }$, the mapping

${\displaystyle B\mapsto \zeta (\omega ,B)\quad (B\in {\mathcal {E}})}$

is a measure on ${\displaystyle (E,{\mathcal {E}})}$

Being locally finite means that the measures

${\displaystyle B\mapsto \zeta (\omega ,B)}$

satisfy ${\displaystyle \zeta (\omega ,{\tilde {B}})<\infty }$ for all bounded measurable sets ${\displaystyle {\tilde {B}}\in {\mathcal {E}}}$ and for all ${\displaystyle \omega \in \Omega }$ except some ${\displaystyle P}$-null set

In the context of stochastic processes there is the related concept of a stochastic kernel, probability kernel, Markov kernel.

As a random element

Define

${\displaystyle {\tilde {\mathcal {M}}}:=\{\mu \mid \mu {\text{ is measure on }}(E,{\mathcal {E}})\}}$

and the subset of locally finite measures by

${\displaystyle {\mathcal {M}}:=\{\mu \in {\tilde {\mathcal {M}}}\mid \mu ({\tilde {B}})<\infty {\text{ for all bounded measurable }}{\tilde {B}}\in {\mathcal {E}}\}}$

For all bounded measurable ${\displaystyle {\tilde {B}}}$, define the mappings

${\displaystyle I_{\tilde {B}}\colon \mu \mapsto \mu ({\tilde {B}})}$

from ${\displaystyle {\tilde {\mathcal {M}}}}$ to ${\displaystyle \mathbb {R} }$. Let ${\displaystyle {\tilde {\mathbb {M} }}}$ be the ${\displaystyle \sigma }$-algebra induced by the mappings ${\displaystyle I_{\tilde {B}}}$ on ${\displaystyle {\tilde {\mathcal {M}}}}$ and ${\displaystyle \mathbb {M} }$ the ${\displaystyle \sigma }$-algebra induced by the mappings ${\displaystyle I_{\tilde {B}}}$ on ${\displaystyle {\mathcal {M}}}$. Note that ${\displaystyle {\tilde {\mathbb {M} }}|_{\mathcal {M}}=\mathbb {M} }$.

A random measure is a random element from ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ to ${\displaystyle ({\tilde {\mathcal {M}}},{\tilde {\mathbb {M} }})}$ that almost surely takes values in ${\displaystyle ({\mathcal {M}},\mathbb {M} )}$^[3][4][5]

Basic related concepts

Intensity measure

For a random measure ${\displaystyle \zeta }$, the measure ${\displaystyle \operatorname {E} \zeta }$ satisfying

${\displaystyle \operatorname {E} \left[\int f(x)\;\zeta (\mathrm {d} x)\right]=\int f(x)\;\operatorname {E} \zeta (\mathrm {d} x)}$

for every positive measurable function ${\displaystyle f}$ is called the intensity measure of ${\displaystyle \zeta }$. The intensity measure exists for every random measure and is a s-finite measure.

Supporting measure

For a random measure ${\displaystyle \zeta }$, the measure ${\displaystyle \nu }$ satisfying

${\displaystyle \int f(x)\;\zeta (\mathrm {d} x)=0{\text{ a.s. }}{\text{ iff }}\int f(x)\;\nu (\mathrm {d} x)=0}$

for all positive measurable functions is called the supporting measure of ${\displaystyle \zeta }$. The supporting measure exists for all random measures and can be chosen to be finite.

Laplace transform

For a random measure ${\displaystyle \zeta }$, the Laplace transform is defined as

${\displaystyle {\mathcal {L}}_{\zeta }(f)=\operatorname {E} \left[\exp \left(-\int f(x)\;\zeta (\mathrm {d} x)\right)\right]}$

for every positive measurable function ${\displaystyle f}$.

Basic properties

Measurability of integrals

For a random measure ${\displaystyle \zeta }$, the integrals

${\displaystyle \int f(x)\zeta (\mathrm {d} x)}$

and ${\displaystyle \zeta (A):=\int \mathbf {1} _{A}(x)\zeta (\mathrm {d} x)}$

for positive ${\displaystyle {\mathcal {E}}}$-measurable ${\displaystyle f}$ are measurable, so they are random variables.

Uniqueness

The distribution of a random measure is uniquely determined by the distributions of

${\displaystyle \int f(x)\zeta (\mathrm {d} x)}$

for all continuous functions with compact support ${\displaystyle f}$ on ${\displaystyle E}$. For a fixed semiring ${\displaystyle {\mathcal {I}}\subset {\mathcal {E}}}$ that generates ${\displaystyle {\mathcal {E}}}$ in the sense that ${\displaystyle \sigma ({\mathcal {I}})={\mathcal {E}}}$, the distribution of a random measure is also uniquely determined by the integral over all positive simple ${\displaystyle {\mathcal {I}}}$-measurable functions ${\displaystyle f}$.^[6]

Decomposition

A measure generally might be decomposed as:

${\displaystyle \mu =\mu _{d}+\mu _{a}=\mu _{d}+\sum _{n=1}^{N}\kappa _{n}\delta _{X_{n}},}$

Here ${\displaystyle \mu _{d}}$ is a diffuse measure without atoms, while ${\displaystyle \mu _{a}}$ is a purely atomic measure.

Random counting measure

A random measure of the form:

${\displaystyle \mu =\sum _{n=1}^{N}\delta _{X_{n}},}$

where ${\displaystyle \delta }$ is the Dirac measure, and ${\displaystyle X_{n}}$ are random variables, is called a point process^[1][2] or random counting measure. This random measure describes the set of N particles, whose locations are given by the (generally vector valued) random variables ${\displaystyle X_{n}}$. The diffuse component ${\displaystyle \mu _{d}}$ is null for a counting measure.

In the formal notation of above a random counting measure is a map from a probability space to the measurable space (${\displaystyle N_{X}}$, ${\displaystyle {\mathfrak {B}}(N_{X})}$) a measurable space. Here ${\displaystyle N_{X}}$ is the space of all boundedly finite integer-valued measures ${\displaystyle N\in M_{X}}$ (called counting measures).

The definitions of expectation measure, Laplace functional, moment measures and stationarity for random measures follow those of

• Poisson random measure
• Vector measure
• Ensemble

References