Mixed binomial process

A mixed binomial process is a special point process in probability theory. They naturally arise from restrictions of 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. 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.

Conditional on, a mixed Binomial processe has the Laplace transform
for any positive, measurable function.

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.

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 random measures include the Poisson random measure, negative binomial random measure, and binomial random measure.