Results in Mathematics

, Volume 72, Issue 1–2, pp 715–730 | Cite as

A Note on the Birkhoff Ergodic Theorem

  • Nikola Sandrić


The classical Birkhoff ergodic theorem states that for an ergodic Markov process the limiting behaviour of the time average of a function (having finite p-th moment, \(p\ge 1\), with respect to the invariant measure) along the trajectories of the process, starting from the invariant measure, is a.s. and in the p-th mean constant and equals to the space average of the function with respect to the invariant measure. The crucial assumption here is that the process starts from the invariant measure, which is not always the case. In this paper, under the assumptions that the underlying process is a Markov process on Polish space, that it admits an invariant probability measure and that its marginal distributions converge to the invariant measure in the \(L^{1}\)-Wasserstein metric, we show that the assertion of the Birkhoff ergodic theorem holds in the p-th mean, \(p\ge 1\), for any bounded Lipschitz function and any initial distribution of the process.


Birkhoff ergodic theorem ergodicity Markov process Wasserstein metric 

Mathematics Subject Classification

60J05 60J25 


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  1. 1.
    Bass, R.F.: Markov processes with Lipschitz semigroups. Trans. Am. Math. Soc. 267(1), 307–320 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bhattacharya, R.N.: On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrscheinlichkeitstheorie Verwandte Geb. 60(2), 185–201 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, New York (1968)zbMATHGoogle Scholar
  5. 5.
    Bogachev, V.I.: Measure Theory, vol. II. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  6. 6.
    Butkovsky, O.: Subgeometric rates of convergence of Markov processes in the Wasserstein metric. Ann. Appl. Probab. 24(2), 526–552 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, M.-F.: From Markov Chains to Non-equilibrium Particle Systems, 2nd edn. World Scientific Publishing Co., Inc., River Edge (2004)CrossRefzbMATHGoogle Scholar
  8. 8.
    Folland, G.B.: Real Analysis. Wiley, New York (1984)zbMATHGoogle Scholar
  9. 9.
    Hairer, M.: Ergodic properties of Markov processes. Lecture notes, University of Warwick. (2006)
  10. 10.
    Hairer, M., Mattingly, J.C., Scheutzow, M.: Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab. Theory Relat. Fields 149(1–2), 223–259 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hernández-Lerma, O., Lasserre, J.B.: On the classification of Markov chains via occupation measures. Appl. Math. (Warsaw) 27(4), 489–498 (2000)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)zbMATHGoogle Scholar
  13. 13.
    Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. Appl. Probab. 25(3), 487–517 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability, second edition edn. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  15. 15.
    Miculescu, R.: Approximations by Lipschitz functions generated by extensions. Real Anal. Exch. 28(1), 33–40 (2002/2003)Google Scholar
  16. 16.
    Schilling, R.L.: Conservativeness and extensions of feller semigroups. Positivity 2, 239–256 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Schilling, R.L., Wang, J.: Strong Feller continuity of Feller processes and semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15(2), 1250010 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tweedie, R.L.: Topological conditions enabling use of Harris methods in discrete and continuous time. Acta Appl. Math. 34(1–2), 175–188 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Villani, C.: Optimal Transport. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar

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© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Civil EngineeringUniversity of ZagrebZagrebCroatia

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