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 $$ P $$ be a probability distribution and let $$ X_i, X_2, \dots $$ be i.i.d. random variables with distribution $$ P $$. Let $$ K $$ be a random variable taking a.s. (almost surely) values in $$ \mathbb N= \{0,1,2, \dots \} $$. Assume that $$ K, X_1, X_2, \dots $$ are independent and let $$ \delta_x $$ denote the Dirac measure on the point $$ x $$.

Then a random measure $$ \xi $$ is called a mixed binomial process iff it has a representation as
 * $$ \xi= \sum_{i=0}^K \delta_{X_i} $$

This is equivalent to $$ \xi $$ conditionally on $$\{ K =n \}$$ being a binomial process based on $$n $$ and $$ P $$.

Laplace transform
Conditional on $$ K=n $$, a mixed Binomial processe has the Laplace transform
 * $$ \mathcal L(f)= \left( \int \exp(-f(x))\; P(\mathrm dx)\right)^n $$

for any positive, measurable function $$ f $$.

Restriction to bounded sets
For a point process $$ \xi $$ and a bounded measurable set $$ B $$ define the restriction of$$ \xi $$ on $$ B $$ as
 * $$ \xi_B(\cdot )= \xi(B \cap \cdot) $$.

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 on
 * $$ P_B(\cdot)= \frac{P(B \cap \cdot)}{P(B)} $$

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.