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 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