Theory of fuzzy non homogeneous Markov systems with fuzzy states

In this paper the theory of fuzzy logic and fuzzy reasoning is combined with the theory of Markov systems and the concept of a fuzzy non homogeneous Markov system with fuzzy states is introduced for the first time. This is an effort to deal with the uncertainty introduced in the estimation of the transition probabilities, especially when social mobility is being measured, and the fact of fuzzy states, in the sense that the categories cannot be precisely measured and are therefore fuzzy. A full description of the methodology is outlined and the basic parameters of the system are provided. Moreover, the expected population structure of the system is estimated in a closed analytic form and the asymptotic behaviour, variability of the state sizes of the system and the rate of convergence of the relative population structure are given. The proposed methodology is illustrated through an example of measuring intergenerational educational mobility.