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

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

Comment

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

Properties

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

\mathcal L(f)=\exp \left(- \int 1-\exp(-f(y))\; (x \mu)(\mathrm dy)\right).

Sources