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Wikipedia / 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 be a separable complete metric space be its \sigma -algebra.)

As a transition kernel

is a (a.s.) locally finite transition kernel from an abstract probability space.

Being a transition kernel means that

Being locally finite means that the measures

-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

and the subset of locally finite measures by

, define the mappings

.

Intensity measure

satisfying

. The intensity measure exists for every random measure and is a s-finite measure.

Supporting measure

satisfying

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

Laplace transform

, the Laplace transform is defined as

.

Basic properties

Measurability of integrals

, the integrals

are measurable, so they are random variables.

Uniqueness

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

. For a fixed semiring, the distribution of a random measure is also uniquely determined by the integral over all positive simple.

Decomposition

A measure generally might be decomposed as:

is a purely atomic measure.

Random counting measure

A random measure of the form:

is the Dirac measure are random variables, is called a point process or random counting measure. This random measure describes the set of N 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 (called counting 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.

See also

References