Summary
For every Radon measure λ on the real line let P λ 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:
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 D qP. For instance we have D q D λ=P q λ. Let II denote the set of all point processes and consequently
It is shown, that the set
of point processes is identical with the set of all doubly stochastic Poisson processes.
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Mecke, J. Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse. Z. Wahrscheinlichkeitstheorie verw Gebiete 11, 74–81 (1968). https://doi.org/10.1007/BF00538387
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DOI: https://doi.org/10.1007/BF00538387