Transforming spatial point processes into Poisson processes
Let \(N\) be a simple point process in the quadrant \({\mathbb R}_+^2\). The author shows that if the `coordinate sectional' intensities (\(\lambda _1(\omega ,s,t),\lambda _2(\omega ,s,t)\)) exist and if with probability one, \(\int \lambda _1(\omega ,s,t)dt=\infty \) for all \(s>0\), then there exists a family of \({\mathcal F}^1\)-predictable sets \(\{ D_z: z\in {\mathbb R}_+^2\}\) such that \(M:z\mapsto N(D_z)\) is a unit rate Poisson process. A possible generalization to higher dimensions is considered as well. These results generalize the original problem solved by Papangelou and Bremaud, whether the real line can be rescaled in such a way that a given point process on \(\mathbb R\) is transformed into a Poisson one.
scientific article; zbMATH DE number 1554310