Invariance of Poisson measures under random transformations

Nicolas Privault

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 4, page 947-972
  • ISSN: 0246-0203

Abstract

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We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai (Probab. Theory Related Fields103 (1995) 409–429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure.

How to cite

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Privault, Nicolas. "Invariance of Poisson measures under random transformations." Annales de l'I.H.P. Probabilités et statistiques 48.4 (2012): 947-972. <http://eudml.org/doc/272011>.

@article{Privault2012,
abstract = {We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai (Probab. Theory Related Fields103 (1995) 409–429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure.},
author = {Privault, Nicolas},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Poisson measures; random transformations; invariance; Skorohod integral; moment identities; Poisson measure; random transformation},
language = {eng},
number = {4},
pages = {947-972},
publisher = {Gauthier-Villars},
title = {Invariance of Poisson measures under random transformations},
url = {http://eudml.org/doc/272011},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Privault, Nicolas
TI - Invariance of Poisson measures under random transformations
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 4
SP - 947
EP - 972
AB - We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai (Probab. Theory Related Fields103 (1995) 409–429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure.
LA - eng
KW - Poisson measures; random transformations; invariance; Skorohod integral; moment identities; Poisson measure; random transformation
UR - http://eudml.org/doc/272011
ER -

References

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