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

Then a random measure [math]\displaystyle{ \xi }[/math] is called a mixed binomial process iff it has a representation as

[math]\displaystyle{ \xi= \sum_{i=0}^K \delta_{X_i} }[/math]

This is equivalent to [math]\displaystyle{ \xi }[/math] conditionally on [math]\displaystyle{ \{ K =n \} }[/math] being a binomial process based on [math]\displaystyle{ n }[/math] and [math]\displaystyle{ P }[/math].

Properties

Laplace transform

Conditional on [math]\displaystyle{ K=n }[/math], a mixed Binomial processe has the Laplace transform

[math]\displaystyle{ \mathcal L(f)= \left( \int \exp(-f(x))\; P(\mathrm dx)\right)^n }[/math]

for any positive, measurable function [math]\displaystyle{ f }[/math].

Restriction to bounded sets

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

[math]\displaystyle{ \xi_B(\cdot )= \xi(B \cap \cdot) }[/math].

Mixed binomial processes are stable under restrictions in the sense that if [math]\displaystyle{ \xi }[/math] is a mixed binomial process based on [math]\displaystyle{ P }[/math] and [math]\displaystyle{ K }[/math], then [math]\displaystyle{ \xi_B }[/math] is a mixed binomial process based on

[math]\displaystyle{ P_B(\cdot)= \frac{P(B \cap \cdot)}{P(B)} }[/math]

and some random variable [math]\displaystyle{ \tilde K }[/math].

Also if [math]\displaystyle{ \xi }[/math] is a Poisson process or a mixed Poisson process, then [math]\displaystyle{ \xi_B }[/math] 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.