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

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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

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

Being a transition kernel means that

  • For any fixed  , the mapping
 
is measurable from   to  
  • For every fixed  , the mapping
 
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

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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]

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Intensity measure

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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

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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

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For a random measure  , the Laplace transform is defined as

 

for every positive measurable function  .

Basic properties

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Measurability of integrals

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For a random measure  , the integrals

 

and  

for positive  -measurable   are measurable, so they are random variables.

Uniqueness

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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

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A measure generally might be decomposed as:

 

Here   is a diffuse measure without atoms, while   is a purely atomic measure.

Random counting measure

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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

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References

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  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