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Efron type identities for stopping sets and Poisson hulls
Abstract
We consider a Poisson process η on a general Borel space with intensity measure λ and a stopping set Z depending on η. Generalizing earlier results we prove a spatial Markov property for Z. In fact we establish this property for the larger class of finitely insertion tolerant random sets, which include pointwise limits of stopping sets. Using the spatial Markov property of η, we derive several distributional identities for the restrictions of η and λ to Z and the complement of Z. An important special case in Euclidean space is the convex hull of a finite Poisson process. In this case our results generalize classical (and also more recent) identities connecting the number of vertices and the volume of the convex hull.
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BibTeX
@misc{last2026efron,
title = {{Efron type identities for stopping sets and Poisson hulls}},
author = {Günter Last and Ilya Molchanov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.06038},
eprint = {2608.06038},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.06038},
}
RIS
TY - UNPB
TI - Efron type identities for stopping sets and Poisson hulls
AU - Günter Last
AU - Ilya Molchanov
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2608.06038
UR - https://arxiv.org/abs/2608.06038
AB - We consider a Poisson process η on a general Borel space with intensity measure λ and a stopping set Z depending on η. Generalizing earlier results we prove a spatial Markov property for Z. In fact we establish this property for the larger class of finitely insertion tolerant random sets, which include pointwise limits of stopping sets. Using the spatial Markov property of η, we derive several distributional identities for the restrictions of η and λ to Z and the complement of Z. An important special case in Euclidean space is the convex hull of a finite Poisson process. In this case our results generalize classical (and also more recent) identities connecting the number of vertices and the volume of the convex hull.
ER -