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Poisson splitting by factors. (English) Zbl 1277.60087
Summary: Given a homogeneous Poisson process on \(\mathbb R^{d}\) with intensity \(\lambda \), 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 \(\lambda \). In particular, this answers a question of K. Ball [Electron. Commun. Probab. 10, 60–69 (2005; Zbl 1110.60050)], 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 similarly is possible 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.

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
37A50 Dynamical systems and their relations with probability theory and stochastic processes
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[1] Angel, O., Holroyd, A. E. and Soo, T. (2011). Deterministic thinning of finite Poisson processes. Proc. Amer. Math. Soc. 139 707-720. · Zbl 1208.60047 · doi:10.1090/S0002-9939-2010-10535-X
[2] Ball, K. (2005). Monotone factors of i.i.d. processes. Israel J. Math. 150 205-227. · Zbl 1146.37010 · doi:10.1007/BF02762380
[3] Ball, K. (2005). Poisson thinning by monotone factors. Electron. Commun. Probab. 10 60-69 (electronic). · Zbl 1110.60050 · eudml:127246
[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. · Zbl 1206.60037
[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. · Zbl 1042.60064 · doi:10.1016/j.anihpb.2003.12.001 · numdam:AIHPB_2004__40_2_141_0 · eudml:77803
[6] Gurel-Gurevich, O. and Peled, R. (2011). Poisson thickening. Israel J. Math. To appear. Available at . · Zbl 1306.60052 · arxiv.org
[7] Holroyd, A. E., Pemantle, R., Peres, Y. and Schramm, O. (2009). Poisson matching. Ann. Inst. Henri Poincaré Probab. Stat. 45 266-287. · Zbl 1175.60012
[8] Holroyd, A. E. and Peres, Y. (2003). Trees and matchings from point processes. Electron. Commun. Probab. 8 17-27 (electronic). · Zbl 1060.60048 · eudml:124514
[9] Holroyd, A. E. and Peres, Y. (2005). Extra heads and invariant allocations. Ann. Probab. 33 31-52. · Zbl 1097.60032 · doi:10.1214/009117904000000603
[10] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis . Cambridge Univ. Press, Cambridge. Corrected reprint of the 1985 original. · Zbl 0704.15002
[11] Jacod, J. (1975). Two dependent Poisson processes whose sum is still a Poisson process. J. Appl. Probab. 12 170-172. · Zbl 0305.60022 · doi:10.2307/3212423
[12] Kallenberg, O. (2002). Foundations of Modern Probability , 2nd ed. Springer, New York. · Zbl 0996.60001
[13] Keane, M. and Smorodinsky, M. (1977). A class of finitary codes. Israel J. Math. 26 352-371. · Zbl 0357.94012 · doi:10.1007/BF03007652
[14] Keane, M. and Smorodinsky, M. (1979). Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. of Math. (2) 109 397-406. · Zbl 0405.28017 · doi:10.2307/1971117
[15] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3 . Oxford Univ. Press, New York. · Zbl 0771.60001
[16] Last, G. and Thorisson, H. (2009). Invariant transports of stationary random measures and mass-stationarity. Ann. Probab. 37 790-813. · Zbl 1176.60036 · doi:10.1214/08-AOP420
[17] Molchanov, I. (2005). Theory of Random Sets . Springer, London. · Zbl 1109.60001
[18] Ornstein, D. S. (1974). Ergodic Theory , Randomness , and Dynamical Systems . Yale Univ. Press, New Haven. · Zbl 0296.28016
[19] Ornstein, D. S. and Weiss, B. (1987). Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 1-141. · Zbl 0637.28015 · doi:10.1007/BF02790325
[20] Petersen, K. (1989). Ergodic Theory. Cambridge Studies in Advanced Mathematics 2 . Cambridge Univ. Press, Cambridge. Corrected reprint of the 1983 original. · Zbl 0676.28008
[21] Rees, E. G. (1983). Notes on Geometry . Springer, Berlin. · Zbl 0498.51001
[22] Reiss, R. D. (1993). A Course on Point Processes . Springer, New York. · Zbl 0771.60037
[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. · Zbl 1132.37005
[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 \Bbb Z d . Adv. in Appl. Probab. 42 69-82. · Zbl 1205.60098 · doi:10.1239/aap/1269611144
[26] Srivastava, S. M. (1998). A Course on Borel Sets. Graduate Texts in Mathematics 180 . Springer, New York. · Zbl 0903.28001
[27] Thorisson, H. (1996). Transforming random elements and shifting random fields. Ann. Probab. 24 2057-2064. · Zbl 0879.60051 · doi:10.1214/aop/1041903217
[28] Thorisson, H. (2000). Coupling , Stationarity , and Regeneration . Springer, New York. · Zbl 0949.60007
[29] Timár, Á. Invariant matchings of exponential tail on coin flips in \Bbb Z d . Preprint. Available at . · arxiv.org
[30] Timár, Á. (2004). Tree and grid factors for general point processes. Electron. Commun. Probab. 9 53-59 (electronic). · Zbl 1060.60050 · eudml:124515
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