A strong limit theorem for the average of ternary functions of Markov chains in bi-infinite random environments

We consider strong limit theorems for Markov chains in bi-infinite random environments. We first give a new proof of a strong limit theorem of Liu and Yang (1995) on the average of functions of non-homogeneous Markov chains by constructing a nonnegative martingale. As corollaries, for a Markov chain in a bi-infinite random environment, we obtain strong limit theorems for the conditional relative entropy and for the number of times that the Markov chain and related processes reach a given point.