Robert Hable, D. Skulj
Coefficients of ergodicity for Markov chains with uncertain parameters
Metrika, vol. 76, no. 1, pp. 107-133
ne 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.
Beitrag (Sammelband oder Tagungsband)
Robert Hable, D. Skulj
Coefficients of ergodicity for imprecise Markov chaines
Proceedings of the Sixth International Symposium on Imprecise Probability: Theories and Applications (ISIPTA'09) [July 14th - 18th 2009, Department of Mathematical Sciences, Durham University, Durham, UK]
Coefficients of ergodicity are an important tool in measuring convergence of Markov chains. We explore possibilities to generalise the concept to imprecise Markov chains. We find that this can be done in at least two different ways, which both have interesting implications in the study of convergence of imprecise Markov chains. Thus we extend the existing definition of the uniform coefficient of ergodicity and define a new so-called weak coefficient of ergodicity. The definition is based on the endowment of a structure of a metric space to the class of imprecise probabilities. We show that this is possible to do in some different ways, which turn out to coincide.