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.

be a locally finite measure on

be a random variable with

is called a mixed Poisson process based on

is a Poisson process on

Mixed Poisson processes are doubly stochastic in the sense that in a first step, the value of the random variable

This value then determines the "second order stochasticity" by increasing or decreasing the original intensity measure

mixed Poisson processes have the intensity measure