The Annals of Probability

Finitary isomorphisms of Poisson point processes

Terry Soo and Amanda Wilkens

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss (J. Anal. Math. 48 (1987) 1–141) proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary.

Article information

Ann. Probab., Volume 47, Number 5 (2019), 3055-3081.

Received: May 2018
Revised: December 2018
First available in Project Euclid: 22 October 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 37A35: Entropy and other invariants, isomorphism, classification 60G10: Stationary processes 60G55: Point processes

Poisson point process finitary isomorphisms


Soo, Terry; Wilkens, Amanda. Finitary isomorphisms of Poisson point processes. Ann. Probab. 47 (2019), no. 5, 3055--3081. doi:10.1214/18-AOP1332.

Export citation


  • [1] Angel, O., Holroyd, A. E. and Soo, T. (2011). Deterministic thinning of finite Poisson processes. Proc. Amer. Math. Soc. 139 707–720.
  • [2] Devroye, L. (1986). Nonuniform Random Variate Generation. Springer, New York.
  • [3] Foreman, M., Rudolph, D. J. and Weiss, B. (2011). The conjugacy problem in ergodic theory. Ann. of Math. (2) 173 1529–1586.
  • [4] Harvey, N., Holroyd, A. E., Peres, Y. and Romik, D. (2007). Universal finitary codes with exponential tails. Proc. Lond. Math. Soc. (3) 94 475–496.
  • [5] Hogg, R., McKean, J. and Craig, A. (2013). Introduction to Mathematical Statistics, 7th ed. Pearson, Boston, MA.
  • [6] Holroyd, A. E., Lyons, R. and Soo, T. (2011). Poisson splitting by factors. Ann. Probab. 39 1938–1982.
  • [7] Kalikow, S. and Weiss, B. (1992). Explicit codes for some infinite entropy Bernoulli shifts. Ann. Probab. 20 397–402.
  • [8] Keane, M. and Smorodinsky, M. (1977). A class of finitary codes. Israel J. Math. 26 352–371.
  • [9] Keane, M. and Smorodinsky, M. (1979). Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. of Math. (2) 109 397–406.
  • [10] Ornstein, D. (1970). Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 337–352.
  • [11] Ornstein, D. (2013). Newton’s laws and coin tossing. Notices Amer. Math. Soc. 60 450–459.
  • [12] Ornstein, D. S. and Weiss, B. (1987). Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 1–141.
  • [13] Rees, E. G. (1983). Notes on Geometry. Universitext. Springer, Berlin.
  • [14] Soo, T. (2018). Finitary isomorphisms of some infinite entropy Bernoulli flows. Israel J. Math. To appear. Available at arXiv:1806.09349.
  • [15] Srivastava, S. M. (1998). A Course on Borel Sets. Graduate Texts in Mathematics 180. Springer, New York.
  • [16] Weiss, B. (1972). The isomorphism problem in ergodic theory. Bull. Amer. Math. Soc. 78 668–684.