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Poisson thickening

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Abstract

Let × be a Poisson point process of intensity λ on the real line. A thickening of it is a (deterministic) measurable function f such that X∪f(X) is a Poisson point process of intensity λ′ where λ′ > λ. An equivariant thickening is a thickening which commutes with all shifts of the line. We show that a thickening exists but an equivariant thickening does not. We prove similar results for thickenings which commute only with integer shifts and in the discrete and multi-dimensional settings. This answers 3 questions of Holroyd, Lyons and Soo.

We briefly consider also a much more general setup in which we ask for the existence of a deterministic coupling satisfying a relation between two probability measures. We present a conjectured sufficient condition for the existence of such couplings.

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Authors and Affiliations

  1. University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Ori Gurel-Gurevich

  2. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Ron Peled

Authors
  1. Ori Gurel-Gurevich
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  2. Ron Peled
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Corresponding author

Correspondence to Ori Gurel-Gurevich.

Additional information

Research of R.P. supported by NSF Grant OISE 0730136.

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Gurel-Gurevich, O., Peled, R. Poisson thickening. Isr. J. Math. 196, 215–234 (2013). https://doi.org/10.1007/s11856-012-0181-2

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  • DOI: https://doi.org/10.1007/s11856-012-0181-2

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