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 {\displaystyle P} be a probability distribution and let X i , X 2 , … {\displaystyle X_{i},X_{2},\dots } be i.i.d. random variables with distribution P {\displaystyle P} . Let K {\displaystyle K} be a random variable taking a.s. (almost surely) values in N = { 0 , 1 , 2 , … } {\displaystyle \mathbb {N} =\{0,1,2,\dots \}} . Assume that K , X 1 , X 2 , … {\displaystyle K,X_{1},X_{2},\dots } are independent and let δ x {\displaystyle \delta _{x}} denote the Dirac measure on the point x {\displaystyle x} .
Then a random measure ξ {\displaystyle \xi } is called a mixed binomial process iff it has a representation as
ξ = ∑ i = 0 K δ X i {\displaystyle \xi =\sum _{i=0}^{K}\delta _{X_{i}}}This is equivalent to ξ {\displaystyle \xi } conditionally on { K = n } {\displaystyle \{K=n\}} being a binomial process based on n {\displaystyle n} and P {\displaystyle P} .1
Properties
Laplace transform
Conditional on K = n {\displaystyle K=n} , a mixed Binomial processe has the Laplace transform
L ( f ) = ( ∫ exp ( − f ( x ) ) P ( d x ) ) n {\displaystyle {\mathcal {L}}(f)=\left(\int \exp(-f(x))\;P(\mathrm {d} x)\right)^{n}}for any positive, measurable function f {\displaystyle f} .
Restriction to bounded sets
For a point process ξ {\displaystyle \xi } and a bounded measurable set B {\displaystyle B} define the restriction of ξ {\displaystyle \xi } on B {\displaystyle B} as
ξ B ( ⋅ ) = ξ ( B ∩ ⋅ ) {\displaystyle \xi _{B}(\cdot )=\xi (B\cap \cdot )} .Mixed binomial processes are stable under restrictions in the sense that if ξ {\displaystyle \xi } is a mixed binomial process based on P {\displaystyle P} and K {\displaystyle K} , then ξ B {\displaystyle \xi _{B}} is a mixed binomial process based on
P B ( ⋅ ) = P ( B ∩ ⋅ ) P ( B ) {\displaystyle P_{B}(\cdot )={\frac {P(B\cap \cdot )}{P(B)}}}and some random variable K ~ {\displaystyle {\tilde {K}}} .
Also if ξ {\displaystyle \xi } is a Poisson process or a mixed Poisson process, then ξ B {\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. 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. 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 //doi.org/10.1002/mma.6224 ↩