Mixed Poisson process

In probability theory, a mixed Poisson process is a special point process that is a generalization of a Poisson process. Mixed Poisson processes are simple example for Cox processes.

Definition
Let $$ \mu $$ be a locally finite measure on $$ S $$ and let $$ X $$ be a random variable with $$ X \geq 0 $$ almost surely.

Then a random measure $$ \xi $$ on $$ S $$ is called a mixed Poisson process based on $$ \mu $$ and $$ X $$ iff $$ \xi $$ conditionally on $$ X=x $$ is a Poisson process on $$ S $$ with intensity measure $$ x\mu $$.

Comment
Mixed Poisson processes are doubly stochastic in the sense that in a first step, the value of the random variable $$ X $$ is determined. This value then determines the "second order stochasticity" by increasing or decreasing the original intensity measure $$ \mu $$.

Properties
Conditional on $$ X=x $$ mixed Poisson processes have the intensity measure $$ x \mu $$ and the Laplace transform
 * $$ \mathcal L(f)=\exp \left(- \int 1-\exp(-f(y))\; (x \mu)(\mathrm dy)\right) $$.