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

Then a random measure \( \xi \) 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 \( \xi \) conditionally on \( {\displaystyle \{K=n\}} \) being a binomial process based on n and 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 f.

Restriction to bounded sets

For a point process \( \xi \) and a bounded measurable setB define the restriction of \( \xi \) on B as

ξ B ( ⋅ ) = ξ ( B ∩ ⋅ ) {\displaystyle \xi _{B}(\cdot )=\xi (B\cap \cdot )} {\displaystyle \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 \( {\displaystyle \xi _{B}} \) 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 \( \xi \)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

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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