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