Journal of Theoretical Probability

, Volume 21, Issue 3, pp 687–703 | Cite as

A Conditional CLT which Fails for Ergodic Components

  • L. Ouchti
  • D. Volný


We show that the conditional central limit theorem can take place for a stationary process defined on a nonergodic dynamical system while this last does not satisfy the central limit theorem for any ergodic component. There exists an ergodic Markov chain such that the conditional central limit theorem is satisfied for an invariant measure but fails to hold for almost all starting points.


Central limit theorem Martingale Ergodic measure Ergodic component Annealed CLT Quenched CLT Markov chain 

Mathematics Subject Classification (2000)

60G42 60F05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dedecker, J., Merlevède, F.: Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30, 1014–1081 (2002) Google Scholar
  2. 2.
    Derriennic, Y., Lin, M.: The central limit theorem for Markov chains with normal transition operators. Probab. Theory Relat. Fields 119, 508–528 (2001) CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    del Junco, A., Rosenblatt, J.: Counterexample in ergodic theory and number theory. Math. Ann. 245, 185–197 (1979) CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Derriennic, Y., Lin, M.: The central limit theorem for random walks on orbits of probability preserving transformations. Contemp. Math. (2007, to appear) Google Scholar
  5. 5.
    Durieu, O., Volný, D.: Comparison between criteria leading to the weak invariance principle. Ann. Inst. H. Pincaré (2007, to appear) Google Scholar
  6. 6.
    Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes. Commun. Math. Phys. 104, 1–19 (1986) CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Maxwell, M., Woodroofe, M.: Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28, 713–724 (2000) CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Volný, D.: On non ergodic versions of limit theorems. Apl. Mat. 5, 351–363 (1987) Google Scholar
  9. 9.
    Volný, D.: Counter examples to the central limit problem for stationary dependent random variables. Yokohama Math. J. 36, 69–78 (1988) MathSciNetGoogle Scholar
  10. 10.
    Woodroofe, M.: A central limit theorem for functions of a Markov chain with application to shifts. Stoch. Process. Appl. 41, 33–44 (1992) CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Wu, W.B., Woodroofe, M.: Martingale approximations for sums of stationary processes. Ann. Probab. 32(2), 1674–1690 (2004) CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.LPA de Beaune BellegardeBeaune la rolandeFrance
  2. 2.Laboratoire de Mathématiques Raphaël Salem UMR CNRS 6085Université de RouenSaint Etienne du RouvrayFrance

Personalised recommendations