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 $\mathbb {N} =\{0,1,2,\dots \}$ . 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 _{i=0}^{K}\delta _{X_{i}}

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

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 {d} x)\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 {P(B\cap \cdot )}{P(B)}}

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.[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

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.
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

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[Kallenberg72-1] 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.

[Kallenberg77-2] 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.

[Caleb_Bastian-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