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.

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

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

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

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

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

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

• Poisson point processes