Poisson splitting by factors

@article{Holroyd2011PoissonSB,
  title={Poisson splitting by factors},
  author={Alexander E. Holroyd and Russell Lyons and Terry Soo},
  journal={Annals of Probability},
  year={2011},
  volume={39},
  pages={1938-1982}
}
Given a homogeneous Poisson process on ℝd with intensity λ, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to λ. In particular, this answers a question of Ball [Electron. Commun. Probab. 10 (2005) 60–69], who proved that in d = 1, the Poisson points may be similarly partitioned (via a… Expand

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References

SHOWING 1-10 OF 53 REFERENCES
Poisson thickening
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′ >Expand
Tree and Grid factors of General Point processes
We study isomorphism invariant point processes of $R^d$ whose groups of symmetries are almost surely trivial. We define a 1-ended, locally finite tree factor on the points of the process, that is, aExpand
Trees and Matchings from Point Processes
A factor graph of a point process is a graph whose vertices are the points of the process, and which is constructed from the process in a deterministic isometry-invariant way. We prove that the dExpand
Poisson Matching
Suppose that red and blue points occur as independent homogeneous Poisson processes in Rd. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. ForExpand
Invariant matchings of exponential tail on coin flips in $\Z^d$
Consider Bernoulli(1/2) percolation on $\Z^d$, and define a perfect matching between open and closed vertices in a way that is a deterministic equivariant function of the configuration. We want toExpand
Deterministic thinning of finite Poisson processes
Let Pi and Gamma be homogeneous Poisson point processes on a fixed set of finite volume. We prove a necessary and sufficient condition on the two intensities for the existence of a coupling of Pi andExpand
A Zero-one Law for Linear Transformations of Levy Noise
A Levy noise on Rd assigns a random real "mass" �(B) to each Borel subset B of R d with finite Lebesgue measure. The distribution of �(B) only depends on the Lebesgue measure of B, and if B1,...,BnExpand
Poisson trees, succession lines and coalescing random walks
Abstract We give a deterministic algorithm to construct a graph with no loops (a tree or a forest) whose vertices are the points of a d-dimensional stationary Poisson process S⊂ R d . The algorithmExpand
Extra heads and invariant allocations
Let n be an ergodic simple point process on E d and let n* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of Π and n*; that is, one canExpand
Transforming random elements and shifting random fields
Consider a locally compact second countable topological transformation group acting measurably on an arbitrary space. We show that the distributions of two random elements X and $X'$ in this spaceExpand
...
1
2
3
4
5
...