On the uniformly ergodic inheritance of finite Hidden Markov Models

This research delves into the inheritance of probabilistic characteristics from the hidden to the observable chain within Hidden Markov Models. Specifically, it identifies the conditions for the observable chain to inherit the Markov property from the hidden chain. The conditions allow for a comprehensive substitution of any finite Markovian dynamics with the mechanisms of Hidden Markov Models. Furthermore, the study explores the inheritance of uniform ergodicity. It also investigates the relationship between the stationary distributions of both chains, and provides a comparative analysis of their geometric convergence rate. The discussion concludes by highlighting two promising applications.