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
:$\mathcal L(f)=\exp \left(- \int 1-\exp(-f(y))\; (x \mu)(\mathrm dy)\right)$.

Sources

*{{cite book , last1=Kallenberg , first1=Olav , author-link1=Olav Kallenberg , year=2017 , title=Random Measures, Theory and Applications, location= Switzerland , publisher=Springer , doi= 10.1007/978-3-319-41598-7, isbn=978-3-319-41596-3

Poisson point processes