Constructing one-step and limiting fuzzy transition probabilities for finite Markov chains

In this paper a methodology for constructing one-step and limiting transition probabilities for fuzzy Markov chains is proposed. This method involves employing Dempster–Shafer type mass functions to construct transition probabilities for set-valued Markov chains in which the sets are subsets of the original state space. The relationship between the composition of mass functions via the Dempster–Shafer rule of combination and set-valued Markov chains is utilized to obtain these transition probabilities. Limiting transition probabilities make use of a limit theorem for the infinite composition of homogeneous mass functions. These set-valued transition probabilities are then converted into transition probabilities on the original state space. Since more than one sequence of mass functions can be under consideration with varying degrees of certitude, the resulting transition probabilities are typically fuzzy sets.