Nu-transform

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.

denote the Dirac measure on the point

be a simple point measure on

This means that for distinct

for every bounded set

be a Markov kernel from

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 a point process

, a second point process

is called a

{ ξ = μ }

, the point process

is a Cox process directed by the random measure

is again a Cox-process, directed by the random measure

ξ ⋅ ν

(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

μ ⋅ ν

, then the Laplace transform of

is given by for all bounded, positive and measurable functions