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 $\mathbb{\mathbb{N}}=\{0,1,2,\dots \}$. 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 $P$.

Properties

Laplace transform

Conditional on $K=n$, a mixed Binomial processe has the Laplace transform

${ℒ}(f)={(\int \exp (-f(x))P(\mathrm{d}x))}^n$

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.