Binomial process

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

Definition

Let [math]\displaystyle{ P }[/math] be a probability distribution and [math]\displaystyle{ n }[/math] be a fixed natural number. Let [math]\displaystyle{ X_1, X_2, \dots, X_n }[/math] be i.i.d. random variables with distribution [math]\displaystyle{ P }[/math], so [math]\displaystyle{ X_i \sim P }[/math] for all [math]\displaystyle{ i \in \{1, 2, \dots, n \} }[/math].

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

[math]\displaystyle{ \xi= \sum_{i=1}^n \delta_{X_i}, }[/math]

where [math]\displaystyle{ \delta_{X_i(A)}=\begin{cases}1, &\text{if }X_i\in A,\\ 0, &\text{otherwise}.\end{cases} }[/math]

Properties

Name

The name of a binomial process is derived from the fact that for all measurable sets [math]\displaystyle{ A }[/math] the random variable [math]\displaystyle{ \xi(A) }[/math] follows a binomial distribution with parameters [math]\displaystyle{ P(A) }[/math] and [math]\displaystyle{ n }[/math]:

[math]\displaystyle{ \xi(A) \sim \operatorname{Bin}(n,P(A)). }[/math]

Laplace-transform

The Laplace transform of a binomial process is given by

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

for all positive measurable functions [math]\displaystyle{ f }[/math].

Intensity measure

The intensity measure [math]\displaystyle{ \operatorname{E}\xi }[/math] of a binomial process [math]\displaystyle{ \xi }[/math] is given by

[math]\displaystyle{ \operatorname{E}\xi =n P. }[/math]

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 [math]\displaystyle{ K }[/math]. Therefore mixed binomial processes conditioned on [math]\displaystyle{ K=n }[/math] are binomial process based on [math]\displaystyle{ n }[/math] and [math]\displaystyle{ P }[/math].

Literature

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