The Annals of Probability

Poisson splitting by factors

Alexander E. Holroyd, Russell Lyons, and Terry Soo

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Abstract

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 translation-equivariant function) so that one set forms a Poisson process of lower intensity, and asked whether the same is possible for all d. We do not know whether it is possible similarly to add points (again chosen as a deterministic function of a Poisson process) to obtain a Poisson process of higher intensity, but we prove that this is not possible under an additional finitariness condition.

Article information

Source
Ann. Probab., Volume 39, Number 5 (2011), 1938-1982.

Dates
First available in Project Euclid: 18 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1318940786

Digital Object Identifier
doi:10.1214/11-AOP651

Mathematical Reviews number (MathSciNet)
MR2884878

Zentralblatt MATH identifier
1277.60087

Subjects
Primary: 60G55: Point processes 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]

Keywords
Poisson process stochastic domination factor map thinning

Citation

Holroyd, Alexander E.; Lyons, Russell; Soo, Terry. Poisson splitting by factors. Ann. Probab. 39 (2011), no. 5, 1938--1982. doi:10.1214/11-AOP651. https://projecteuclid.org/euclid.aop/1318940786


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References

  • [1] Angel, O., Holroyd, A. E. and Soo, T. (2011). Deterministic thinning of finite Poisson processes. Proc. Amer. Math. Soc. 139 707–720.
  • [2] Ball, K. (2005). Monotone factors of i.i.d. processes. Israel J. Math. 150 205–227.
  • [3] Ball, K. (2005). Poisson thinning by monotone factors. Electron. Commun. Probab. 10 60–69 (electronic).
  • [4] Evans, S. N. (2010). A zero–one law for linear transformations of Lévy noise. In Algebraic Methods in Statistics and Probability II (M. A. Viana and H. P. Wynn, eds.). Contemporary Mathematics 516 189–197. Amer. Math. Soc., Providence, RI.
  • [5] Ferrari, P. A., Landim, C. and Thorisson, H. (2004). Poisson trees, succession lines and coalescing random walks. Ann. Inst. Henri Poincaré Probab. Stat. 40 141–152.
  • [6] Gurel-Gurevich, O. and Peled, R. (2011). Poisson thickening. Israel J. Math. To appear. Available at arXiv:0911.5377.
  • [7] Holroyd, A. E., Pemantle, R., Peres, Y. and Schramm, O. (2009). Poisson matching. Ann. Inst. Henri Poincaré Probab. Stat. 45 266–287.
  • [8] Holroyd, A. E. and Peres, Y. (2003). Trees and matchings from point processes. Electron. Commun. Probab. 8 17–27 (electronic).
  • [9] Holroyd, A. E. and Peres, Y. (2005). Extra heads and invariant allocations. Ann. Probab. 33 31–52.
  • [10] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis. Cambridge Univ. Press, Cambridge. Corrected reprint of the 1985 original.
  • [11] Jacod, J. (1975). Two dependent Poisson processes whose sum is still a Poisson process. J. Appl. Probab. 12 170–172.
  • [12] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [13] Keane, M. and Smorodinsky, M. (1977). A class of finitary codes. Israel J. Math. 26 352–371.
  • [14] Keane, M. and Smorodinsky, M. (1979). Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. of Math. (2) 109 397–406.
  • [15] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. Oxford Univ. Press, New York.
  • [16] Last, G. and Thorisson, H. (2009). Invariant transports of stationary random measures and mass-stationarity. Ann. Probab. 37 790–813.
  • [17] Molchanov, I. (2005). Theory of Random Sets. Springer, London.
  • [18] Ornstein, D. S. (1974). Ergodic Theory, Randomness, and Dynamical Systems. Yale Univ. Press, New Haven.
  • [19] Ornstein, D. S. and Weiss, B. (1987). Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 1–141.
  • [20] Petersen, K. (1989). Ergodic Theory. Cambridge Studies in Advanced Mathematics 2. Cambridge Univ. Press, Cambridge. Corrected reprint of the 1983 original.
  • [21] Rees, E. G. (1983). Notes on Geometry. Springer, Berlin.
  • [22] Reiss, R. D. (1993). A Course on Point Processes. Springer, New York.
  • [23] Serafin, J. (2006). Finitary codes, a short survey. In Dynamics & Stochastics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 48 262–273. IMS, Beachwood, OH.
  • [24] Sinaĭ, J. G. (1962). A weak isomorphism of transformations with invariant measure. Dokl. Akad. Nauk SSSR 147 797–800.
  • [25] Soo, T. (2010). Translation-equivariant matchings of coin flips on ℤd. Adv. in Appl. Probab. 42 69–82.
  • [26] Srivastava, S. M. (1998). A Course on Borel Sets. Graduate Texts in Mathematics 180. Springer, New York.
  • [27] Thorisson, H. (1996). Transforming random elements and shifting random fields. Ann. Probab. 24 2057–2064.
  • [28] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.
  • [29] Timár, Á. Invariant matchings of exponential tail on coin flips in ℤd. Preprint. Available at arXiv:0909.1090.
  • [30] Timár, Á. (2004). Tree and grid factors for general point processes. Electron. Commun. Probab. 9 53–59 (electronic).