Nu-transform explained

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

For measures

Let

\delta _{x} denote the Dirac measure on the point x and let \mu be a simple point measure on S . This means that\mu = \sum _{k} \delta _{s _{k}}

for distinct

s _{k} \in S and \mu(B)< inf ty for every bounded set B in S . Further, let \nu be a Markov kernel from S to T .

Let

\tau _{k} be independent random elements with distribution \nu _{s _{k}} = \nu(s _{k, ⋅ )} . Then the point process\zeta = \sum _{k} \delta _{\tau _{k}}

is called the ν-transform of the measure

\mu if it is locally finite, meaning that \zeta(B) < inf ty for every bounded set B

For point processes

\xi , a second point process \zeta is called a \nu -transform of \xi if, conditional on \{ \xi = \mu\} , the point process \zeta is a \nu -transform of \mu .

Properties

Stability

If

\zeta is a Cox process directed by the random measure \xi , then the \nu -transform of \zeta is again a Cox-process, directed by the random measure \xi ⋅ \nu (see Transition kernel#Composition of kernels)

Therefore, the

\nu -transform of a Poisson process with intensity measure \mu is a Cox process directed by a random measure with distribution \mu ⋅ \nu .

Laplace transform

It

\zeta is a \nu -transform of \xi , then the Laplace transform of \zeta is given byl L _{\zeta} (f) = \exp \left( \int log \left[ \int \exp( - f(t)) \mu _{s(dt)\right]\xi(ds)\right)}

for all bounded, positive and measurable functions

f .

References

^{[1] [2]}

Notes and References

1. Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 10.1007/978-3-319-41598-7. 978-3-319-41596-3. 73.
2. Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 10.1007/978-3-319-41598-7. 978-3-319-41596-3. 75.