Mixed binomial process

When mixed binomial processes is called a special class of point processes in the theory of stochastic processes , a branch of probability theory . Mixed binomial processes are generalizations of binomial processes in the sense that they do not consider a deterministic number of random variables , but a random one.

definition

Given a measurement space and independent, identically distributed random variables with values ​​in . Furthermore, let us be another random variable that is independent of all and almost certainly takes on values ​​in . Let it denote the Dirac measure on the point , so ${\displaystyle (S,{\mathcal {A}})}$ ${\displaystyle X_{1},X_{2},X_{3},\dots }$ ${\displaystyle S}$ ${\displaystyle Y}$ ${\displaystyle X_{i}}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle \delta _{x}}$ ${\displaystyle x}$

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

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

Then that's through

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

defined random measure on a mixed binomial process. Is the distribution of , so , it means also by and mixed given binomial process. ${\displaystyle \zeta }$ ${\displaystyle (S,{\mathcal {A}})}$ ${\displaystyle P}$ ${\displaystyle X_{i}}$ ${\displaystyle X_{i}\sim P}$ ${\displaystyle \zeta }$ ${\displaystyle Y}$ ${\displaystyle P}$

properties

Intensity measure and distribution

For every measurable set there is a binomially distributed random variable with parameters and . So it applies ${\displaystyle A}$ ${\displaystyle \zeta (A)}$ ${\displaystyle Y}$ ${\displaystyle P(A)}$

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

If and are integrable, then according to the formula of Wald ${\displaystyle \operatorname {E} (Y)<\infty }$ ${\displaystyle X_{i}}$

${\displaystyle \operatorname {E} (\zeta (A))=\operatorname {E} (Y)\operatorname {E} (\delta _{X_{i}}(A))=\operatorname {E} (Y)P(A)}$.

Here again a random measure can be seen. Thus the intensity measure of a mixed binomial process is through in this case ${\displaystyle \delta _{X_{i}}}$ ${\displaystyle \operatorname {E\zeta } }$ ${\displaystyle \zeta }$

${\displaystyle \operatorname {E\zeta } =\operatorname {E} (Y)P}$

given.

Relationship to the binomial process

If the random variable almost certainly assumes the value , the mixed binomial process changes into a binomial process , which is determined by and the distribution of . ${\displaystyle Y}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle X_{i}}$

Laplace transform

The Laplace transform of a mixed binomial process is given by ${\displaystyle Y=k}$

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

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

literature

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