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Concept
Random measure
Summary
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.
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 )
A random measure is a (a.s.) locally finite transition kernel from a (abstract) probability space to .
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.
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
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.
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.
For a random measure , the Laplace transform is defined as
for every positive measurable function .
For a random measure , the integrals
and
for positive -measurable are measurable, so they are random variables.
The distribution of a random measure is uniquely determined by the distributions of
for all continuous functions with compact support on .
Official source
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