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Moderate deviations for Markov chains with atom

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  • Djellout, H.
  • Guillin, A.

Abstract

We obtain in this paper moderate deviations for functional empirical processes of general state space valued Markov chains with atom under weak conditions: a tail condition on the first time of return to the atom, and usual conditions on the class of functions. Our proofs rely on the regeneration method and sharp conditions issued of moderate deviations of independent random variables. We prove our result in the nonseparable case for additive and unbounded functionals of Markov chains, extending the work of de Acosta and Chen (J. Theoret. Probab. (1998) 75-110) and Wu (Ann. Probab. (1995) 420-445). One may regard it as the analog for the Markov chains of the beautiful characterization of moderate deviations for i.i.d. case of Ledoux 1992. Some applications to Markov chains with a countable state space are considered.

Suggested Citation

  • Djellout, H. & Guillin, A., 2001. "Moderate deviations for Markov chains with atom," Stochastic Processes and their Applications, Elsevier, vol. 95(2), pages 203-217, October.
    • Handle: RePEc:eee:spapps:v:95:y:2001:i:2:p:203-217
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    References listed on IDEAS

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    1. Tsai, Tsung-Hsi, 2000. "Empirical law of the iterated logarithm for Markov chains with a countable state space," Stochastic Processes and their Applications, Elsevier, vol. 89(2), pages 175-191, October.
    2. Guillin, Arnaud, 2001. "Moderate deviations of inhomogeneous functionals of Markov processes and application to averaging," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 287-313, April.
    3. Levental, Shlomo, 1990. "Uniform CLT for Markov chains with a countable state space," Stochastic Processes and their Applications, Elsevier, vol. 34(2), pages 245-253, April.
    4. Nummelin, Esa & Tuominen, Pekka, 1982. "Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 12(2), pages 187-202, March.
    5. Chen, Xia, 1997. "Moderate deviations for m-dependent random variables with Banach space values," Statistics & Probability Letters, Elsevier, vol. 35(2), pages 123-134, September.
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    Cited by:

    1. Eva Löcherbach & Dasha Loukianova, 2012. "Deviation Inequalities for Centered Additive Functionals of Recurrent Harris Processes Having General State Space," Journal of Theoretical Probability, Springer, vol. 25(1), pages 231-261, March.

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