Binomial process

As binomial processes is called a special class of point processes in the theory of stochastic processes , a branch of probability theory . Binomial processes are similar to Poisson processes , but the number of events per interval is binomially distributed and not Poisson distributed .

definition

Given an integer and a probability distribution on a measurement space as well as independent, identical and randomly distributed random variables . So it applies to everyone . Furthermore, denote the Dirac measure on the point , i.e. ${\displaystyle n}$ ${\displaystyle P}$ ${\displaystyle (X,{\mathcal {B}})}$ ${\displaystyle n}$ ${\displaystyle P}$ ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$ ${\displaystyle X_{i}\sim P}$ ${\displaystyle i}$ ${\displaystyle \delta _{x}}$ ${\displaystyle x}$

${\displaystyle \delta _{x}(A):={\begin{cases}1&{\text{ falls }}x\in A\ \\0&{\text{ sonst }}\end{cases}}}$

for . ${\displaystyle A\in {\mathcal {B}}}$

Then that's through

${\displaystyle \zeta :=\sum _{i=1}^{n}\delta _{X_{i}}}$

defined random measure on a binomial process. ${\displaystyle \zeta }$ ${\displaystyle (X,{\mathcal {B}})}$

comment

For every measurable amount applies by definition ${\displaystyle A}$

${\displaystyle \zeta (A)=\#\{i\mid X_{i}\in A\}}$

Here denotes the power of the set , i.e. the number of its elements. The process thus counts how many of the random variables take on values ​​in the set . Thus, for every measurable set, the random variable is always binomially distributed with parameters and , so it is true ${\displaystyle \#M}$ ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle \zeta (A)}$ ${\displaystyle n}$ ${\displaystyle P(A)}$

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

properties

Associated jump process

In the real case, i.e. for , the jump process belonging to the point process is given by ${\displaystyle (X,{\mathcal {B}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

${\displaystyle X_{t}:=\zeta ((-\infty ,t])=\#\{i\mid X_{i}\leq t\}}$.

It indicates how many of the random variables take on values ​​less than or equal. ${\displaystyle t}$

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 measurable positive functions . ${\displaystyle f}$

Intensity measure

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

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

Generalizations

A generalization of the binomial processes are mixed binomial processes . The number of random variables, which is deterministic in binomial processes, is replaced by a random variable. ${\displaystyle n}$

literature

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