Coefficients of ergodicity for Markov chains with uncertain parameters
The authors explore the coefficients of ergodicity for the case of imprecise Markov chains. They extend one of the most commonly used coefficients of ergodicity to the case of imprecise Markov chains. The main idea behind the coefficients of ergodicity is to measure the distances between rows of transition matrices, which in the precise case are probability distributions and in the imprecise case are, similarly, imprecise probability distibutions. The main contribution of the paper is the introduction of a new so-called weak coefficient of ergodicity that, in an alternative way, characterizes the type of convergence already studied by other authors. They show that the conditions for convergence provided by this coefficient are necessary and sufficient if convex sets of probabilities are used to describe the imprecision of Markov chains, and the independence condition for rows of transition operators is assumed. The paper is conceived in 6 sections. After the Introduction, in Section 2, they review some theory on lower expectation operators that form a basis for the model of imprecise probabilities with the structure of a metric space. In Section 3, the possibilities to endow the family of imprecise probabilities with a structure of a metric space are explored. Then, the model of imprecise Markov chains that are used is described in Section 4. In the next section, they give an alternative characterisation of the uniform coefficient of ergodicity and, as the main contribution of the paper, the authors define a new so-called weak coefficient of ergodicity which provides necessary and sufficient conditions for convergence for an imprecise Markov chain and measures its rate. A numerical example is given, too. Some conclusions are presented in the last section. Here, the authors emphasize again the main contribution of the paper. This is a good and instructive paper.
scientific article; zbMATH DE number 6138687
scientific article