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 δ x {\displaystyle \delta _{x}} denote the Dirac measure on the point x {\displaystyle x} and let μ {\displaystyle \mu } be a simple point measure on S {\displaystyle S} . This means that
μ = ∑ k δ s k {\displaystyle \mu =\sum _{k}\delta _{s_{k}}}
for distinct s k ∈ S {\displaystyle s_{k}\in S} and μ ( B ) < ∞ {\displaystyle \mu (B)<\infty } for every bounded set B {\displaystyle B} in S {\displaystyle S} . Further, let ν {\displaystyle \nu } be a Markov kernel from S {\displaystyle S} to T {\displaystyle T} . Let τ k {\displaystyle \tau _{k}} be independent random elements with distribution ν s k = ν ( s k , ⋅ ) {\displaystyle \nu _{s_{k}}=\nu (s_{k},\cdot )} . Then the point process
ζ = ∑ k δ τ k {\displaystyle \zeta =\sum _{k}\delta _{\tau _{k}}}
is called the ν-transform of the measure μ {\displaystyle \mu } if it is locally finite, meaning that ζ ( B ) < ∞ {\displaystyle \zeta (B)<\infty } for every bounded set B {\displaystyle B}
For point processes For a point process ξ {\displaystyle \xi } , a second point process ζ {\displaystyle \zeta } is called a ν {\displaystyle \nu } -transform of ξ {\displaystyle \xi } if, conditional on { ξ = μ } {\displaystyle \{\xi =\mu \}} , the point process ζ {\displaystyle \zeta } is a ν {\displaystyle \nu } -transform of μ {\displaystyle \mu } .
Properties
Stability If ζ {\displaystyle \zeta } is a Cox process directed by the random measure ξ {\displaystyle \xi } , then the ν {\displaystyle \nu } -transform of ζ {\displaystyle \zeta } is again a Cox-process, directed by the random measure ξ ⋅ ν {\displaystyle \xi \cdot \nu } (see Transition kernel#Composition of kernels) Therefore, the ν {\displaystyle \nu } -transform of a Poisson process with intensity measure μ {\displaystyle \mu } is a Cox process directed by a random measure with distribution μ ⋅ ν {\displaystyle \mu \cdot \nu } .
Laplace transform It ζ {\displaystyle \zeta } is a ν {\displaystyle \nu } -transform of ξ {\displaystyle \xi } , then the Laplace transform of ζ {\displaystyle \zeta } is given by
L ζ ( f ) = exp ( ∫ log [ ∫ exp ( − f ( t ) ) μ s ( d t ) ] ξ ( d s ) ) {\displaystyle {\mathcal {L}}_{\zeta }(f)=\exp \left(\int \log \left[\int \exp(-f(t))\mu _{s}(\mathrm {d} t)\right]\xi (\mathrm {d} s)\right)}
for all bounded, positive and measurable functions f {\displaystyle f} .
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