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Nu-transform

From Wikipedia, the free encyclopedia

In the theory of stochastic processes, a ν-transform is an operation that transforms a measure or a point process into a different point process. Intuitively the ν-transform randomly relocates the points of the point process, with the type of relocation being dependent on the position of each point.

Definition

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For measures

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Let denote the Dirac measure on the point and let be a simple point measure on . This means that

for distinct and for every bounded set in . Further, let be a Markov kernel from to .

Let be independent random elements with distribution . Then the point process

is called the ν-transform of the measure if it is locally finite, meaning that for every bounded set [1]

For point processes

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For a point process , a second point process is called a -transform of if, conditional on , the point process is a -transform of .[1]

Properties

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Stability

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If is a Cox process directed by the random measure , then the -transform of is again a Cox-process, directed by the random measure (see Transition kernel#Composition of kernels)[2]

Therefore, the -transform of a Poisson process with intensity measure is a Cox process directed by a random measure with distribution .

Laplace transform

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It is a -transform of , then the Laplace transform of is given by

for all bounded, positive and measurable functions .[1]

References

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  1. ^ a b c Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 73. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
  2. ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 75. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.