Mixed binomial process
A mixed binomial process is a special point process in probability theory. They naturally arise from restrictions of (mixed) Poisson processes bounded intervals.
Definition
Let be a probability distribution and let be i.i.d. random variables with distribution . Let be a random variable taking a.s. (almost surely) values in . Assume that are independent and let denote the Dirac measure on the point .
Then a random measure is called a mixed binomial process iff it has a representation as
This is equivalent to conditionally on being a binomial process based on and .[1]
Properties
Laplace transform
Conditional on , a mixed Binomial processe has got the Laplace transform
for any positive, measurable function .
Restriction to bounded sets
For a point process and a bounded measurable set define the restriction of on as
- .
Mixed binomial processes are stable under restrictions in the sense that if is a mixed binomial process based on and , then is a mixed binomial process based on
and some random variable .
Also if is a Poisson process or a mixed Poisson process, then is a mixed binomial process.[2]
References
- ↑ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 72. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
- ↑ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 77. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.