Binomial process

Let $P$ be a probability distribution and $n$ be a fixed natural number. Let $X\_1,\; X\_2,\; \backslash dots,\; X\_n$ be i.i.d. random variables with distribution $P$, so $X\_i\; \backslash sim\; P$ for all $i\; \backslash in\; \backslash \{1,\; 2,\; \backslash dots,\; n\; \backslash \}$.

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

: $\backslash xi=\; \backslash sum\_\{i=1\}^n\; \backslash delta\_\{X\_i\},$

where

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

:$\backslash xi(A)\; \backslash sim\; \backslash operatorname\{Bin\}(n,P(A)).$

The Laplace transform of a binomial process is given by

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

for all positive measurable functions $f$.

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

:$\backslash operatorname\{E\}\backslash xi\; =n\; P.$

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 $P$.

• {{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}}

Category:Point processes