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Coefficients of ergodicity for Markov chains with uncertain parameters

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.

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Correspondence to D. Škulj.

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An earlier version of this paper, Škulj and Hable (2009), was presented at the ISIPTA’09 conference in Durham, UK, 2009.

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Škulj, D., Hable, R. Coefficients of ergodicity for Markov chains with uncertain parameters. Metrika 76, 107–133 (2013). https://doi.org/10.1007/s00184-011-0378-0

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  • DOI: https://doi.org/10.1007/s00184-011-0378-0

Keywords

  • Markov chain
  • Imprecise Markov chain
  • Coefficient of ergodicity
  • Weak coefficient of ergodicity
  • Uniform coefficient of ergodicity
  • Lower expectation
  • Upper expectation
  • Hausdorff metric
  • Convergence of Markov chains