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 $$ i \in \{1, 2, \dots, n \}$$.

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


 * $$ \xi= \sum_{i=1}^n \delta_{X_i}, $$

where $$\delta_{X_i(A)}=\begin{cases}1, &\text{if }X_i\in A,\\ 0, &\text{otherwise}.\end{cases}$$

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

for all positive measurable functions $$ f $$.

Intensity measure
The 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 $$ P $$.