The Annals of Probability

A monotone Sinai theorem

Anthony Quas and Terry Soo

Full-text: Open access

Abstract

Sinai proved that a nonatomic ergodic measure-preserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy and is stochastically dominated by the original measure can be obtained as a monotone factor; that is, the factor map has the property that for each point in the domain, its image under the factor map is coordinatewise smaller than or equal to the original point.

Article information

Source
Ann. Probab., Volume 44, Number 1 (2016), 107-130.

Dates
Received: November 2013
Revised: September 2014
First available in Project Euclid: 2 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1454423036

Digital Object Identifier
doi:10.1214/14-AOP968

Mathematical Reviews number (MathSciNet)
MR3456333

Zentralblatt MATH identifier
1359.37014

Subjects
Primary: 37A35: Entropy and other invariants, isomorphism, classification 60G10: Stationary processes 60E15: Inequalities; stochastic orderings

Keywords
Sinai factor theorem stochastic domination monotone coupling Burton–Rothstein

Citation

Quas, Anthony; Soo, Terry. A monotone Sinai theorem. Ann. Probab. 44 (2016), no. 1, 107--130. doi:10.1214/14-AOP968. https://projecteuclid.org/euclid.aop/1454423036


Export citation

References

  • [1] Akcoglu, M. A., del Junco, A. and Rahe, M. (1979). Finitary codes between Markov processes. Z. Wahrsch. Verw. Gebiete 47 305–314.
  • [2] Angel, O., Holroyd, A. E. and Soo, T. (2011). Deterministic thinning of finite Poisson processes. Proc. Amer. Math. Soc. 139 707–720.
  • [3] Ball, K. (2005). Monotone factors of i.i.d. processes. Israel J. Math. 150 205–227.
  • [4] Ball, K. (2005). Poisson thinning by monotone factors. Electron. Commun. Probab. 10 60–69 (electronic).
  • [5] Blum, J. R. and Hanson, D. L. (1963). On the isomorphism problem for Bernoulli schemes. Bull. Amer. Math. Soc. 69 221–223.
  • [6] Breiman, L. (1957). The individual ergodic theorem of information theory. Ann. Math. Statist. 28 809–811.
  • [7] Burton, R. and Rothstein, A. (1977). Isomorphism theorems in ergodic theory. Technical report, Oregon State Univ., Corvallis, OR.
  • [8] Burton, R. M., Keane, M. S. and Serafin, J. (2000). Residuality of dynamical morphisms. Colloq. Math. 84/85 307–317.
  • [9] Cornfeld, I. P., Fomin, S. V. and Sinaĭ, Ya. G. (1982). Ergodic Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 245. Springer, New York. Translated from the Russian by A. B. Sosinskiĭ.
  • [10] del Junco, A. (1981). Finitary codes between one-sided Bernoulli shifts. Ergodic Theory Dynam. Systems 1 285–301.
  • [11] del Junco, A. (1990). Bernoulli shifts of the same entropy are finitarily and unilaterally isomorphic. Ergodic Theory Dynam. Systems 10 687–715.
  • [12] Denker, M. and Keane, M. (1979). Almost topological dynamical systems. Israel J. Math. 34 139–160.
  • [13] de la Rue, T. (2006). An introduction to joinings in ergodic theory. Discrete Contin. Dyn. Syst. 15 121–142.
  • [14] Downarowicz, T. (2011). Entropy in Dynamical Systems. New Mathematical Monographs 18. Cambridge Univ. Press, Cambridge.
  • [15] Dudley, R. (1989). Real Analysis and Probability. Wadsworth, Pacific Grove, CA.
  • [16] Evans, S. N. (2010). A zero–one law for linear transformations of Lévy noise. In Algebraic Methods in Statistics and Probability II. Contemp. Math. 516 189–197. Amer. Math. Soc., Providence, RI.
  • [17] Gurel-Gurevich, O. and Peled, R. (2013). Poisson thickening. Israel J. Math. 196 215–234.
  • [18] Hall, P. (1935). On representatives of subsets. J. London Math. Soc.(1) 10 26–30.
  • [19] Halmos, P. R. (1961). Recent progress in ergodic theory. Bull. Amer. Math. Soc. 67 70–80.
  • [20] 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.
  • [21] Holroyd, A. E., Lyons, R. and Soo, T. (2011). Poisson splitting by factors. Ann. Probab. 39 1938–1982.
  • [22] Katok, A. (2007). Fifty years of entropy in dynamics: 1958–2007. J. Mod. Dyn. 1 545–596.
  • [23] Keane, M. and Smorodinsky, M. (1977). A class of finitary codes. Israel J. Math. 26 352–371.
  • [24] Keane, M. and Smorodinsky, M. (1979). Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. of Math. (2) 109 397–406.
  • [25] Krieger, W. (1970). On entropy and generators of measure-preserving transformations. Trans. Amer. Math. Soc. 149 453–464.
  • [26] Mešalkin, L. D. (1959). A case of isomorphism of Bernoulli schemes. Dokl. Akad. Nauk SSSR 128 41–44.
  • [27] Mester, P. (2013). Invariant monotone coupling need not exist. Ann. Probab. 41 1180–1190.
  • [28] Meyerovitch, T. (2013). Ergodicity of Poisson products and applications. Ann. Probab. 41 3181–3200.
  • [29] Ornstein, D. (1970). Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 337–352.
  • [30] Ornstein, D. (2013). Newton’s laws and coin tossing. Notices Amer. Math. Soc. 60 450–459.
  • [31] Ornstein, D. S. (1974). Ergodic Theory, Randomness, and Dynamical Systems. Yale Mathematical Monographs 5. Yale Univ. Press, New Haven, CT.
  • [32] Ornstein, D. S. and Weiss, B. (1975). Unilateral codings of Bernoulli systems. Israel J. Math. 21 159–166.
  • [33] Ornstein, D. S. and Weiss, B. (1987). Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 1–141.
  • [34] Petersen, K. (1989). Ergodic Theory. Cambridge Studies in Advanced Mathematics 2. Cambridge Univ. Press, Cambridge. Corrected reprint of the 1983 original.
  • [35] Propp, J. G. (1991). Coding Markov chains from the past. Israel J. Math. 75 289–328.
  • [36] Quas, A. and Soo, T. (2014). Ergodic universality of some topological dynamical systems. Trans. Amer. Math. Soc. To appear. Available at http://arxiv.org/abs/1208.3501.
  • [37] Rudolph, D. J. (1981). A characterization of those processes finitarily isomorphic to a Bernoulli shift. In Ergodic Theory and Dynamical Systems, I (College Park, Md., 19791980). Progr. Math. 10 1–64. Birkhäuser, Boston, MA.
  • [38] Rudolph, D. J. (1990). Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces. Clarendon, New York.
  • [39] 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.
  • [40] Shea, S. (2013). Finitarily Bernoulli factors are dense. Fund. Math. 223 49–54.
  • [41] Shea, S. M. (2012). On the marker method for constructing finitary isomorphisms. Rocky Mountain J. Math. 42 293–304.
  • [42] Sinaĭ, Ja. G. (1964). On a weak isomorphism of transformations with invariant measure. Mat. Sb. (N.S.) 63 (105) 23–42.
  • [43] Sinai, Y. G. (2010). Selecta. Volume I. Ergodic Theory and Dynamical Systems. Springer, New York.
  • [44] Srivastava, S. M. (1998). A Course on Borel Sets. Graduate Texts in Mathematics 180. Springer, New York.
  • [45] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36 423–439.
  • [46] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.
  • [47] Weiss, B. (1972). The isomorphism problem in ergodic theory. Bull. Amer. Math. Soc. 78 668–684.