A Conditional CLT which Fails for Ergodic Components
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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.
KeywordsCentral limit theorem Martingale Ergodic measure Ergodic component Annealed CLT Quenched CLT Markov chain
Mathematics Subject Classification (2000)60G42 60F05
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- 1.Dedecker, J., Merlevède, F.: Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30, 1014–1081 (2002) Google Scholar
- 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.Durieu, O., Volný, D.: Comparison between criteria leading to the weak invariance principle. Ann. Inst. H. Pincaré (2007, to appear) Google Scholar
- 8.Volný, D.: On non ergodic versions of limit theorems. Apl. Mat. 5, 351–363 (1987) Google Scholar