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 be a locally finite measure on and let be a random variable with almost surely.

Then a random measure on is called a mixed Poisson process based on and iff conditionally on is a Poisson process on with intensity measure .

Comment

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

Properties

Conditional on mixed Poisson processes have the intensity measure and the Laplace transform

.

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.
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