Mixed Poisson process explained

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

Sources

• Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 10.1007/978-3-319-41598-7. 978-3-319-41596-3.