LAUSR

ABSTRACT Consider a Markov chain with finite alphabets. In this paper, we study the asymptotic properties of moving average, harmonic mean, and strong deviation theorems (limit theorems expressed by inequalities) of moving geometric average of random transition probabilities and the generalized entropy ergodic theorem for Markov chains in single infinite Markovian environments. It is shown that, under some mild conditions, the sequence of the generalized relative entropy density converges almost surely and in . The trick of the proofs is the construction of random variables with a parameter and the application of Borel–Cantelli lemma.