Chernoff bound for channels with infinite memory

The problem of Chernoff bounding is analyzed for symmetric channels with input alphabet of arbitrary finite size or erasure channels with no crossover errors whose memory may be described in terms of the noise sequences by an infinite ergodic irreducible aperiodic Markov chain. Necessary and sufficient conditions are described which guarantee the existence of an exponential upper bound to the probability of error for block coding. These conditions are given in terms of the transition matrix of the Markov chain. It will be observed that, although their application involves merely an inspection of the transition matrix, they are based on the topology of the Markov chain and the convergence of sequences of transition probabilities.