In probability theory, a random measure is a measure-valued random element.[1][2] 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. (The most common example of a separable complete metric space is )

As a transition kernel

A random measure is a (a.s.) locally finite transition kernel from a (abstract) probability space to .[3]

Being a transition kernel means that

is measurable from to
is a measure on

Being locally finite means that the measures

satisfy for all bounded measurable sets and for all except some -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

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 [3][4][5]

Basic related concepts

Intensity measure

Main article: 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 .[6]

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[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 . 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 (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.[7]

See also

References

  1. ^ a b Kallenberg, O., Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986). ISBN 0-12-394960-2 MR854102. An authoritative but rather difficult reference.
  2. ^ a b Jan Grandell, Point processes and random measures, Advances in Applied Probability 9 (1977) 502-526. MR0478331 JSTOR A nice and clear introduction.
  3. ^ a b Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 1. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
  4. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 526. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  5. ^ Daley, D. J.; Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Probability and its Applications. doi:10.1007/b97277. ISBN 0-387-95541-0.
  6. ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 52. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
  7. ^ "Crisan, D., Particle Filters: A Theoretical Perspective, in Sequential Monte Carlo in Practice, Doucet, A., de Freitas, N. and Gordon, N. (Eds), Springer, 2001, ISBN 0-387-95146-6