# Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse

• J. Mecke
Article

## Summary

For every Radon measure λ on the real line let Pλ denote the Poisson process with intensity λ, i.e. with the property that the mean of the occurrences in the Borel set B is λ(B). A point process P is called a doubly stochastic Poisson process, if it can be represented as a mixture of Poisson processes:
$$P = \int {P_2 Q(d\lambda )} ,$$

where Q is a probability measure on a suitable σ-algebra of subsets of the set of all Radon measures (Cox, Bartlett, Kingman).

If the occurrences of a point process P are independently selected with probability q, we obtain a resulting point process DqP. For instance we have DqDλ=Pqλ. Let II denote the set of all point processes and consequently
$$D_q \Pi = \{ D_q P:P \in \Pi \} .$$
It is shown, that the set
$$\bigcap\limits_{0 < q < 1} {D_q } \Pi$$
of point processes is identical with the set of all doubly stochastic Poisson processes.

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