The strong laws of large numbers for hidden Markov model with general state space

In this article, we will study the strong law of large numbers for non homogeneous hidden Markov models, where the observable chain takes values on a general state space. First, we introduce the definition of non homogeneous hidden Markov models. Then we prove some equivalent properties of these hidden Markov models and study some lemmas that are the basis of the main results. Finally, we establish the strong law of large numbers for non homogeneous hidden Markov models.