mixed Poisson process

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

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

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 $\backslash mu$.

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

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

• {{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017 |title=Random Measures, Theory and Applications|location= Switzerland |publisher=Springer |doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3}}

Category:Poisson point processes