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
[edit]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
[edit]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
[edit]Conditional on mixed Poisson processes have the intensity measure
and the Laplace transform
.
Sources
[edit]- Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.