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 be a probability distribution and let be i.i.d. random variables with distribution . Let be a random variable taking a.s. (almost surely) values in . Assume that are independent and let denote the Dirac measure on the point .

Then a random measure is called a mixed binomial process iff it has a representation as

This is equivalent to conditionally on being a binomial process based on and .[1]

Properties

Laplace transform

Conditional on , a mixed Binomial processe has the Laplace transform

for any positive, measurable function .

Restriction to bounded sets

For a point process and a bounded measurable set define the restriction of on as

.

Mixed binomial processes are stable under restrictions in the sense that if is a mixed binomial process based on and , then is a mixed binomial process based on

and some random variable .

Also if is a Poisson process or a mixed Poisson process, then 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].

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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
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