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.
Definition
For measures
Let [math]\displaystyle{ \delta_x }[/math] denote the Dirac measure on the point [math]\displaystyle{ x }[/math] and let [math]\displaystyle{ \mu }[/math] be a simple point measure on [math]\displaystyle{ S }[/math]. This means that
- [math]\displaystyle{ \mu= \sum_k \delta_{s_k} }[/math]
for distinct [math]\displaystyle{ s_k \in S }[/math] and [math]\displaystyle{ \mu(B)\lt \infty }[/math] for every bounded set [math]\displaystyle{ B }[/math] in [math]\displaystyle{ S }[/math]. Further, let [math]\displaystyle{ \nu }[/math] be a Markov kernel from [math]\displaystyle{ S }[/math] to [math]\displaystyle{ T }[/math].
Let [math]\displaystyle{ \tau_k }[/math] be independent random elements with distribution [math]\displaystyle{ \nu_{s_k}=\nu(s_k,\cdot) }[/math]. Then the point process
- [math]\displaystyle{ \zeta = \sum_{k} \delta_{\tau_k} }[/math]
is called the ν-transform of the measure [math]\displaystyle{ \mu }[/math] if it is locally finite, meaning that [math]\displaystyle{ \zeta(B) \lt \infty }[/math] for every bounded set [math]\displaystyle{ B }[/math][1]
For point processes
For a point process [math]\displaystyle{ \xi }[/math], a second point process [math]\displaystyle{ \zeta }[/math] is called a [math]\displaystyle{ \nu }[/math]-transform of [math]\displaystyle{ \xi }[/math] if, conditional on [math]\displaystyle{ \{ \xi=\mu\} }[/math], the point process [math]\displaystyle{ \zeta }[/math] is a [math]\displaystyle{ \nu }[/math]-transform of [math]\displaystyle{ \mu }[/math].[1]
Properties
Stability
If [math]\displaystyle{ \zeta }[/math] is a Cox process directed by the random measure [math]\displaystyle{ \xi }[/math], then the [math]\displaystyle{ \nu }[/math]-transform of [math]\displaystyle{ \zeta }[/math] is again a Cox-process, directed by the random measure [math]\displaystyle{ \xi \cdot \nu }[/math] (see Transition kernel)[2]
Therefore, the [math]\displaystyle{ \nu }[/math]-transform of a Poisson process with intensity measure [math]\displaystyle{ \mu }[/math] is a Cox process directed by a random measure with distribution [math]\displaystyle{ \mu \cdot \nu }[/math].
Laplace transform
It [math]\displaystyle{ \zeta }[/math] is a [math]\displaystyle{ \nu }[/math]-transform of [math]\displaystyle{ \xi }[/math], then the Laplace transform of [math]\displaystyle{ \zeta }[/math] is given by
- [math]\displaystyle{ \mathcal L_{\zeta}(f)= \exp \left( \int \log \left[ \int \exp(-f(t)) \mu_s(\mathrm dt)\right] \xi(\mathrm ds)\right) }[/math]
for all bounded, positive and measurable functions [math]\displaystyle{ f }[/math].[1]
References
- ↑ 1.0 1.1 1.2 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.
- ↑ 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.
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