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 <math> \mu </math> be a locally finite measure on <math> S </math> and let <math> X </math> be a random variable with <math> X \geq 0 </math> almost surely.
Then a random measure <math> \xi </math> on <math> S </math> is called a mixed Poisson process based on <math> \mu </math> and <math> X </math> iff <math> \xi </math> conditionally on <math> X=x </math> is a Poisson process on <math> S </math> with intensity measure <math> x\mu </math>.
== Comment == Mixed Poisson processes are doubly stochastic in the sense that in a first step, the value of the random variable <math> X </math> is determined. This value then determines the "second order stochasticity" by increasing or decreasing the original intensity measure <math> \mu </math>.
== Properties == Conditional on <math> X=x </math> mixed Poisson processes have the intensity measure <math> x \mu </math> and the Laplace transform :<math> \mathcal L(f)=\exp \left(- \int 1-\exp(-f(y))\; (x \mu)(\mathrm dy)\right) </math>.
== Sources == *{{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