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 $\mu$ be a locally finite measure on $S$ and let $X$ be a random variable with $X\ge 0$ 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

$ℒ(f)=\exp (-\int 1-\exp (-f(y))(x\mu )(\mathrm{d}y))$.