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

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

Comment

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

Properties

Conditional on ${\displaystyle X=x}$ mixed Poisson processes have the intensity measure ${\displaystyle x\mu }$ and the Laplace transform

${\displaystyle {\mathcal {L}}(f)=\exp \left(-\int 1-\exp(-f(x))\;(x\mu )(\mathrm {d} x)\right)}$.

Sources

• Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.