The Generalized Entropy Ergodic Theorem with Two Types of Convergence for M-th-Order Nonhomogeneous Hidden Markov Models

A nonhomogeneous hidden Markov model (NHMM) consists of an unobservable nonhomogeneous Markov chain and an observable stochastic process. If the Markov assumption is changed to the condition that each hidden state in each step is related to the previous m states, then the extended model is called an m-th-order NHMM. Considering the calculation of entropy rate in the case of delayed averages, the generalized entropy ergodic theorem with almost-everywhere and \(\mathcal {L}_1\) ${\mathcal{L}}_1$ convergence for m-th-order NHMMs is studied in this paper. The establishment of some inequalities supports the proof of convergence for two types, and the obtained results maintain strict consistency with the corresponding classical results.