mixed binomial process

Let $P$ be a probability distribution and let $X\_i,\; X\_2,\; \backslash dots$ be i.i.d. random variables with distribution $P$. Let $K$ be a random variable taking a.s. (almost surely) values in $\backslash mathbb\; N=\; \backslash \{0,1,2,\; \backslash dots\; \backslash \}$. Assume that $K,\; X\_1,\; X\_2,\; \backslash dots$ are independent and let $\backslash delta\_x$ denote the Dirac measure on the point $x$.

Then a random measure $\backslash xi$ is called a mixed binomial process iff it has a representation as

:$\backslash xi=\; \backslash sum\_\{i=0\}^K\; \backslash delta\_\{X\_i\}$

This is equivalent to $\backslash xi$ conditionally on $\backslash \{\; K\; =n\; \backslash \}$ being a binomial process based on $n$ and $P$.

Conditional on $K=n$, a mixed Binomial processe has the Laplace transform

:$\backslash mathcal\; L(f)=\; \backslash left(\; \backslash int\; \backslash exp(-f(x))\backslash ;\; P(\backslash mathrm\; dx)\backslash right)^n$

for any positive, measurable function $f$.

For a point process $\backslash xi$ and a bounded measurable set $B$ define the restriction of$\backslash xi$ on $B$ as

:$\backslash xi\_B(\backslash cdot\; )=\; \backslash xi(B\; \backslash cap\; \backslash cdot)$.

Mixed binomial processes are stable under restrictions in the sense that if $\backslash xi$ is a mixed binomial process based on $P$ and $K$, then $\backslash xi\_B$ is a mixed binomial process based on

:$P\_B(\backslash cdot)=\; \backslash frac\{P(B\; \backslash cap\; \backslash cdot)\}\{P(B)\}$

and some random variable $\backslash tilde\; K$.

Also if $\backslash xi$ is a Poisson process or a mixed Poisson process, then $\backslash xi\_B$ is a mixed binomial process.

== 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.

{{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}}

{{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}}

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

Category:Point processes