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Coefficients of ergodicity for Markov chains with uncertain parameters

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  • D. Škulj
  • R. Hable

Abstract

One of the central considerations in the theory of Markov chains is their convergence to an equilibrium. Coefficients of ergodicity provide an efficient method for such an analysis. Besides giving sufficient and sometimes necessary conditions for convergence, they additionally measure its rate. In this paper we explore coefficients of ergodicity for the case of imprecise Markov chains. The latter provide a convenient way of modelling dynamical systems where parameters are not determined precisely. In such cases a tool for measuring the rate of convergence is even more important than in the case of precisely determined Markov chains, since most of the existing methods of estimating the limit distributions are iterative. We define a new coefficient of ergodicity that provides necessary and sufficient conditions for convergence of the most commonly used class of imprecise Markov chains. This so-called weak coefficient of ergodicity is defined through an endowment of the structure of a metric space to the class of imprecise probabilities. Therefore we first make a detailed analysis of the metric properties of imprecise probabilities. Copyright Springer-Verlag 2013

Suggested Citation

  • D. Škulj & R. Hable, 2013. "Coefficients of ergodicity for Markov chains with uncertain parameters," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(1), pages 107-133, January.
    • Handle: RePEc:spr:metrik:v:76:y:2013:i:1:p:107-133
    DOI: 10.1007/s00184-011-0378-0
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    References listed on IDEAS

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    1. Jay K. Satia & Roy E. Lave, 1973. "Markovian Decision Processes with Uncertain Transition Probabilities," Operations Research, INFORMS, vol. 21(3), pages 728-740, June.
    2. Arnab Nilim & Laurent El Ghaoui, 2005. "Robust Control of Markov Decision Processes with Uncertain Transition Matrices," Operations Research, INFORMS, vol. 53(5), pages 780-798, October.
    3. Chelsea C. White & Hany K. Eldeib, 1994. "Markov Decision Processes with Imprecise Transition Probabilities," Operations Research, INFORMS, vol. 42(4), pages 739-749, August.
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    1. Stavros Lopatatzidis & Jasper Bock & Gert Cooman & Stijn Vuyst & Joris Walraevens, 2016. "Robust queueing theory: an initial study using imprecise probabilities," Queueing Systems: Theory and Applications, Springer, vol. 82(1), pages 75-101, February.

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