Binomial Process

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 $X_1, X_2, \dots, X_n$ be i.i.d. random variables with distribution $P$, so $X_i \sim P$ for all $i \in \$.

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

: $\xi= \sum_^n \delta_,$
where $\delta_=\begin1, &\textX_i\in A,\\ 0, &\text.\end$

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 $n$:

:$\xi(A) \sim \operatorname(n,P(A)).$

Laplace-transform

The Laplace transform of a binomial process is given by
: \mathcal L_(f)= \left \int \exp(-f(x)) \mathrm P(dx) \rightn

for all positive measurable functions $f$.

Intensity measure

The intensity measure $\operatorname\xi$ of a binomial process $\xi$ is given by

:$\operatorname\xi =n P.$

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

*{{cite book , last1=Kallenberg , first1=Olav , author-link1=Olav Kallenberg , year=2017 , title=Random Measures, Theory and Applications, location= Switzerland , publisher=Springer , doi= 10.1007/978-3-319-41598-7, isbn=978-3-319-41596-3

Point processes