Binomial process

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

Definition

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

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

${\displaystyle \xi =\sum _{i=1}^{n}\delta _{X_{i}},}$

where ${\displaystyle \delta _{X_{i}(A)}={\begin{cases}1,&{\text{if }}X_{i}\in A,\\0,&{\text{otherwise}}.\end{cases}}}$

Properties

Name

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

${\displaystyle \xi (A)\sim \operatorname {Bin} (n,P(A)).}$

Laplace-transform

The Laplace transform of a binomial process is given by

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

for all positive measurable functions ${\displaystyle f}$.

Intensity measure

The intensity measure ${\displaystyle \operatorname {E} \xi }$ of a binomial process ${\displaystyle \xi }$ is given by

${\displaystyle \operatorname {E} \xi =nP.}$

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 ${\displaystyle K}$. Therefore mixed binomial processes conditioned on ${\displaystyle K=n}$ are binomial process based on ${\displaystyle n}$ and ${\displaystyle P}$.

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.