Mixed binomial process explained

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,} ... be i.i.d. random variables with distribution P . Let K be a random variable taking a.s. (almost surely) values in N= \{ 0 , 1 , 2 , ... \} . Assume that K, X _{1,} X _{2,} ... are independent and let \delta _{x} denote the Dirac measure on the point x .\xi is called a mixed binomial process iff it has a representation as\xi = \sum ^{K} _{i =0} \delta _{X _{i}}

This is equivalent to

\xi conditionally on \{ K = n \} being a binomial process based on n and P .

Properties

Laplace transform

Conditional on

K = n , a mixed Binomial processe has the Laplace transforml L(f) = \left( \int \exp( - f(x)) P( dx)\right) ^{n} 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( ⋅} ) = \xi(B \cap ⋅ ) .

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 onP _{B( ⋅ )=} (P(B \cap ⋅ ))/(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.

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

^{[1] [2] [3]}

Notes and References

1. Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 10.1007/978-3-319-41598-7. 978-3-319-41596-3. 72.
2. Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 10.1007/978-3-319-41598-7. 978-3-319-41596-3. 77.
3. Caleb Bastian, Gregory Rempala. Throwing stones and collecting bones: Looking for Poisson-like random measures, Mathematical Methods in the Applied Sciences, 2020.