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 $\mu (B)<\mathrm{\infty }$ 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 $\zeta (B)<\mathrm{\infty }$ 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

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

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