Random measure


In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes.

Definition

Random measures can be defined as transition kernels or as random elements. Both definitions are equivalent. For the definitions, let be a separable complete metric space and let be its Borel -algebra.

As a transition kernel

A random measure is a locally finite transition kernel from a probability space to.
Being a transition kernel means that
Being locally finite means that the measures
satisfy for all bounded measurable sets
and for all except some -null set

As a random element

Define
and the subset of locally finite measures by
For all bounded measurable, define the mappings
from to. Let be the -algebra induced by the mappings on and the -algebra induced by the mappings on. Note that.
A random measure is a random element from to that almost surely takes values in

Basic related concepts

Intensity measure

For a random measure, the measure satisfying
for every positive measurable function is called the intensity measure of. The intensity measure exists for every random measure and is a s-finite measure.

Supporting measure

For a random measure, the measure satisfying
for all positive measurable functions is called the supporting measure of. The supporting measure exists for all random measures and can be chosen to be finite.

Laplace transform

For a random measure, the Laplace transform is defined as
for every positive measurable function.

Basic properties

Measurability of integrals

For a random measure, the integrals
and
for positive -measurable are measurable, so they are random variables.

Uniqueness

The distribution of a random measure is uniquely determined by the distributions of
for all continuous functions with compact support on. For a fixed semiring that generates in the sense that, the distribution of a random measure is also uniquely determined by the integral over all positive simple -measurable functions.

Decomposition

A measure generally might be decomposed as:
Here is a diffuse measure without atoms, while is a purely atomic measure.

Random counting measure

A random measure of the form:
where is the Dirac measure, and are random variables, is called a point process or random counting measure. This random measure describes the set of N particles, whose locations are given by the random variables. The diffuse component 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 a measurable space. Here is the space of all boundedly finite integer-valued measures .
The definitions of expectation measure, Laplace functional, moment measures and stationarity for random measures follow those of point processes. Random measures are useful in the description and analysis of Monte Carlo methods, such as Monte Carlo numerical quadrature and particle filters.