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 $\delta _{x}$ 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 $\mu (B)<\infty$ 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 $\nu _{s_{k}}=\nu (s_{k},\cdot )$. 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 $\zeta (B)<\infty$ for every bounded set $B$

For point processes

For a point process $\xi$, 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 $\xi \cdot \nu$ (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 $\mu \cdot \nu$.

Laplace transform

It $\zeta$ is a $\nu$-transform of $\xi$, then the Laplace transform of $\zeta$ is given by

${\mathcal {L}}_{\zeta }(f)=\exp \left(\int \log \left[\int \exp(-f(t))\mu _{s}(\mathrm {d} t)\right]\xi (\mathrm {d} s)\right)$

for all bounded, positive and measurable functions $f$.