Mixed binomial process

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 ${\displaystyle f}$.

For a point process ${\displaystyle \xi }$ and a bounded measurable set ${\displaystyle B}$ define the restriction of${\displaystyle \xi }$ on ${\displaystyle B}$ as

${\displaystyle \xi _{B}(\cdot )=\xi (B\cap \cdot )}$.

Mixed binomial processes are stable under restrictions in the sense that if ${\displaystyle \xi }$ is a mixed binomial process based on ${\displaystyle P}$ and ${\displaystyle K}$, 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 ${\displaystyle \xi }$ is a Poisson process or a mixed Poisson process, then ${\displaystyle \xi _{B}}$ is a mixed binomial process.^[2]

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]