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 $P$ be a probability distribution and let $X_i, X_2, \dots$ be i.i.d. random variables with distribution $P$. Let $K$ be a random variable taking a.s. (almost surely) values in $\mathbb N= \$. Assume that $K, X_1, X_2, \dots$ are independent and let $\delta_x$ denote the Dirac measure on the point $x$.

Then a random measure $\xi$ is called a mixed binomial process iff it has a representation as
:$\xi= \sum_^K \delta_$

This is equivalent to $\xi$ conditionally on $\$ being a binomial process based on $n$ and $P$.

Properties

Laplace transform

Conditional on $K=n$, a mixed Binomial processe has the Laplace transform
:$\mathcal L(f)= \left( \int \exp(-f(x))\; P(\mathrm dx)\right)^n$

for any positive, measurable function $f$.

Restriction to bounded sets

For a point process $\xi$ and a bounded measurable set $B$ define the restriction of$\xi$ on $B$ as
:$\xi_B(\cdot )= \xi(B \cap \cdot)$.

Mixed binomial processes are stable under restrictions in the sense that if $\xi$ is a mixed binomial process based on $P$ and $K$, then $\xi_B$ is a mixed binomial process based on
:$P_B(\cdot)= \frac$

and some random variable $\tilde K$.

Also if $\xi$ is a Poisson process or a mixed Poisson process, then $\xi_B$ 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.

References

{{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

Point processes