“Nu-transform”的意思、由来-开放百科全书

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Nu-transform

释义

Definition
For measures For point processes
Properties
Stability Laplace transform
References

In the theory of stochastic processes, a ν-transform is a 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

For measures

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

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

Stability

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)^[3]

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

Laplace transform

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

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

References

^[1]^[2]

1 : Point processes

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