At the heart of ergodic theory are the ergodic theorems: results providing sufficient conditions for dynamical systems or random processes to possess ergodic properties, that is, for sample averages of the form
$$ < f > _n = \tfrac{1}{n}\sum\limits_{i = 0}^{n - 1} {fT^i } $$
to converge to an invariant limit. Traditional developments of the pointwise ergodic theorem focus on stationary systems and use a subsidiary result known as the maximal ergodic lemma (or theorem) to prove the ergodic theorem. The general result for AMS systems then follows since an AMS source inherits ergodic properties from its stationary mean; that is, since the set {x: < f > n (x) converges } is invariant and since a system and its stationary mean place equal probability on all invariant sets, one will possess almost everywhere ergodic properties with respect to a class of measurements if and only if the other one does and the limiting sample averages will be the same.


Ergodic Theorem Invariant Function Prob Ability Ergodic Measure Ergodic Property 
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  1. 1.
    G. D. Birkhoff, “Proof of the ergodic theorem,” Proc. Nat. Acad. Sci., vol. 17, pp. 656–660, 1931.CrossRefGoogle Scholar
  2. 2.
    A. Garsia, “A simple proof of E. Hoph’s maximal ergodic theorem,” J. Math, Mech., vol. 14, pp. 381–2, 1965.MathSciNetzbMATHGoogle Scholar
  3. 3.
    R. M. Gray and L. D. Davisson, “The ergodic decomposition of discrete stationary random processes,” IEEE Trans. on Info. Theory, vol. IT-20, pp. 625–636, September 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    J. M. Hammersley and D. J. A. Welsh, “First-passage, percolation, subadditive processes, stochastic networks, and generalized renewal theory,” Bernoulli-Bayes-Laplace Anniversary Volume, Springer, Berlin, 1965.Google Scholar
  5. 5.
    E. Hoph, Ergodentheorie, Springer-Verlag, Berlin, 1937.CrossRefGoogle Scholar
  6. 6.
    R. Jones, “New proof for the maximal ergodic theorem and the Hardy-Littlewood maximal inequality,” Proc. AMS, vol. 87, pp. 681–684, 1983.zbMATHCrossRefGoogle Scholar
  7. 7.
    I. Katznelson and B. Weiss, “A simple proof of some ergodic theorems,” Israel Journal of Mathematics, vol. 42, pp. 291–296, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    J. F. C. Kingman, “The ergodic theory of subadditive stochastic processes,” Ann. Probab., vol. 1, pp. 883–909, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    N. Kryloff and N. Bogoliouboff, “La théorie générale de la mesure dans son application ä 1’étude des systèmes de la mecanique non linéaire,” Ann. of Math., vol. 38, pp. 65–113, 1937.MathSciNetCrossRefGoogle Scholar
  10. 10.
    J. Nedoma, “On the ergodicity and r-ergodicity of stationary probability measures,” Zeitschrift Wahrscheinlichkeitstheorie, vol. 2, pp. 90–97, 1963.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    D. Ornstein and B. Weiss, “The Shannon-McMillan-Breiman theorem for a class of amenable groups,” Israel J. of Math, vol. 44, pp. 53–60, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J. C. Oxtoby, “Ergodic Sets,” Bull Amer. Math. Soc, vol. Volume 58, pp. 116–136, 1952.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    V. A. Rohlin, “Selected topics from the metric theory of dynamical systems,” Uspechi Mat. Nauk., vol. 4, pp. 57–120, 1949. AMS Trans. (2)49MathSciNetGoogle Scholar
  14. 14.
    P. C. Shields, “The ergodic and entropy theorems revisited,” IEEE Transactions on Information Theory, vol. IT-33, pp. 263–266, March 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    J. von NeumAnn. “Zur operatorenmethode in der klassischen mechanik,” Ann. of Math., vol. 33, pp. 587–642, 1932.MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media New York 1988

Authors and Affiliations

  • Robert M. Gray
    • 1
  1. 1.Department of Electrical EngineeringStanford UniversityStanfordUSA

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