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

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

Comment

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

Properties

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

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

Sources

  • Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. doi:. ISBN978-3-319-41596-3.