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 <math> P </math> be a probability distribution and let <math> X_i, X_2, \dots </math> be i.i.d. random variables with distribution <math> P </math>. Let <math> K </math> be a random variable taking a.s. (almost surely) values in <math> \mathbb N= \{0,1,2, \dots \} </math>. Assume that <math> K, X_1, X_2, \dots </math> are independent and let <math> \delta_x </math> denote the Dirac measure on the point <math> x </math>.
Then a random measure <math> \xi </math> is called a mixed binomial process iff it has a representation as :<math> \xi= \sum_{i=0}^K \delta_{X_i} </math>
This is equivalent to <math> \xi </math> conditionally on <math>\{ K =n \}</math> being a binomial process based on <math>n </math> and <math> P </math>.<ref name="Kallenberg72"/>
== Properties == === Laplace transform === Conditional on <math> K=n </math>, a mixed Binomial processe has the Laplace transform :<math> \mathcal L(f)= \left( \int \exp(-f(x))\; P(\mathrm dx)\right)^n </math>
for any positive, measurable function <math> f </math>.
=== Restriction to bounded sets === For a point process <math> \xi </math> and a bounded measurable set <math> B </math> define the restriction of<math> \xi </math> on <math> B </math> as :<math> \xi_B(\cdot )= \xi(B \cap \cdot) </math>.
Mixed binomial processes are stable under restrictions in the sense that if <math> \xi </math> is a mixed binomial process based on <math> P </math> and <math> K </math>, then <math> \xi_B </math> is a mixed binomial process based on :<math> P_B(\cdot)= \frac{P(B \cap \cdot)}{P(B)} </math>
and some random variable <math> \tilde K </math>.
Also if <math> \xi </math> is a Poisson process or a mixed Poisson process, then <math> \xi_B </math> is a mixed binomial process.<ref name="Kallenberg77"/>
== Examples ==
Poisson-type random measures are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under thinning, that are examples of mixed binomial processes. They are the only distributions in the canonical non-negative power series family of distributions to possess this property and include the Poisson distribution, negative binomial distribution, and binomial distribution. Poisson-type (PT) random measures include the Poisson random measure, negative binomial random measure, and binomial random measure.<ref name="Caleb Bastian" />
== References == <references> <ref name="Kallenberg72"> {{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|pages=72}} </ref> <ref name="Kallenberg77"> {{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|pages=77}} </ref> <ref name="Caleb Bastian">Caleb Bastian, Gregory Rempala. Throwing stones and collecting bones: Looking for Poisson-like random measures, Mathematical Methods in the Applied Sciences, 2020. doi:10.1002/mma.6224</ref> </references>
Category:Point processes