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

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

${\displaystyle \xi =\sum _{i=0}^{K}\delta _{X_{i}}}$

This is equivalent to ${\displaystyle \xi }$ conditionally on ${\displaystyle \{K=n\}}$ being a binomial process based on ${\displaystyle n}$ and ${\displaystyle P}$.^[1]

Properties

Laplace transform

Conditional on ${\displaystyle K=n}$, a mixed Binomial processe has the Laplace transform

${\displaystyle {\mathcal {L}}(f)=\left(\int \exp(-f(x))\;P(\mathrm {d} x)\right)^{n}}$

for any positive, measurable function ${\displaystyle f}$.

Restriction to bounded sets

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

${\displaystyle \xi _{B}(\cdot )=\xi (B\cap \cdot )}$.

Mixed binomial processes are stable under restrictions in the sense that if ${\displaystyle \xi }$ is a mixed binomial process based on ${\displaystyle P}$ and ${\displaystyle K}$, then ${\displaystyle \xi _{B}}$ is a mixed binomial process based on

${\displaystyle P_{B}(\cdot )={\frac {P(B\cap \cdot )}{P(B)}}}$

and some random variable ${\displaystyle {\tilde {K}}}$.

Also if ${\displaystyle \xi }$ is a Poisson process or a mixed Poisson process, then ${\displaystyle \xi _{B}}$ is a mixed binomial process.^[2]

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.^[3]

References

1. 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.
2. 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.
3. 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