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On the ergodic decomposition for a class of Markov chains

Author

Listed:
  • Costa, O.L.V.
  • Dufour, F.

Abstract

In this paper we present sufficient conditions for the Doeblin decomposition, and necessary and sufficient conditions for an ergodic decomposition for a Markov chain satisfying a T'-condition, which is a condition adapted from the paper (Statist. and Probab. Lett. 50 (2000) 13). Under no separability assumption on the [sigma]-field, it is shown that the T'-condition is sufficient for the condition that there are no uncountable disjoint absorbing sets and, under some hypothesis, it is also necessary. For the case in which the [sigma]-field is countable generated and separated, this condition is equivalent to the existence of a T continuous component for the Markov chain. Furthermore, under the assumption that the space is a compact separable metric space, it is shown that the Foster-Lyapunov criterion is necessary and sufficient for the existence of an invariant probability measure for the Markov chain, and that every probability measure for the Markov chain is, in this case, non-singular.

Suggested Citation

  • Costa, O.L.V. & Dufour, F., 2005. "On the ergodic decomposition for a class of Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 115(3), pages 401-415, March.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:3:p:401-415
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    References listed on IDEAS

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    1. Tweedie, R. L., 2001. "Drift conditions and invariant measures for Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 345-354, April.
    2. Costa, O. L. V. & Dufour, F., 2000. "Invariant probability measures for a class of Feller Markov chains," Statistics & Probability Letters, Elsevier, vol. 50(1), pages 13-21, October.
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    Cited by:

    1. Frank H. Page & Myrna H. Wooders, 2009. "Endogenous Network Dynamics," Working Papers 2009.28, Fondazione Eni Enrico Mattei.
    2. Fu, Jing & Page, Frank & Zigrand, Jean-Pierre, 2022. "Layered networks, equilibrium dynamics, and stable coalitions," LSE Research Online Documents on Economics 118874, London School of Economics and Political Science, LSE Library.
    3. Gong, Rui & Page, Frank & Wooders, Myrna, 2015. "Endogenous correlated network dynamics," LSE Research Online Documents on Economics 65098, London School of Economics and Political Science, LSE Library.
    4. Gong, Rui & Page, Frank, 2016. "Systemic risk and the dynamics of temporary financial networks," LSE Research Online Documents on Economics 67810, London School of Economics and Political Science, LSE Library.
    5. Jing Fu & Frank Page & Jean-Pierre Zigrand, 2023. "Correction to: Layered Networks, Equilibrium Dynamics, and Stable Coalitions," Dynamic Games and Applications, Springer, vol. 13(2), pages 669-704, June.
    6. Jing Fu & Frank Page & Jean-Pierre Zigrand, 2023. "Layered Networks, Equilibrium Dynamics, and Stable Coalitions," Dynamic Games and Applications, Springer, vol. 13(2), pages 636-668, June.

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