Binomial process explained

A binomial process is a special point process in probability theory.

Definition

Let

P be a probability distribution and n be a fixed natural number. Let X _{1,} X _{2,} ... , X _{n} be i.i.d. random variables with distribution P , so X _{i} \sim P for all i \in \{ 1 , 2 , ... , n \} .

Then the binomial process based on n and P is the random measure

\xi = \sum ^{n} _{i =1} \delta _{X _{i}} , where \delta _{X _{i(A)}} = \begin{cases} 1 , & if X _{i\in} A,\ 0 , & otherwise .\end{cases}

Properties

Name

The name of a binomial process is derived from the fact that for all measurable sets

A the random variable \xi(A) follows a binomial distribution with parameters P(A) and n :\xi(A) \sim \operatorname{Bin}(n,P(A)).

Laplace-transform

The Laplace transform of a binomial process is given by

l L _{P,n} (f) = \left[ \int \exp( - f(x)) P(dx) \right] ^{n}

for all positive measurable functions

f .

Intensity measure

\operatorname{E}\xi of a binomial process \xi is given by\operatorname{E}\xi = n P.

Generalizations

A generalization of binomial processes are mixed binomial processes. In these point processes, the number of points is not deterministic like it is with binomial processes, but is determined by a random variable

K . Therefore mixed binomial processes conditioned on K = n are binomial process based on n and P .

Literature

• Book: Kallenberg . Olav . Olav Kallenberg . 2017 . Random Measures, Theory and Applications. Switzerland . Springer . 10.1007/978-3-319-41598-7. 978-3-319-41596-3.