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

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

Comment

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

Properties

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

[math]\displaystyle{ \mathcal L(f)=\exp \left(- \int 1-\exp(-f(y))\; (x \mu)(\mathrm dy)\right) }[/math].

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.