  As a class of state-dependent channels, Markov channels have been long
studied in information theory for characterizing the feedback capacity and
error exponent. This paper studies a more general variant of such channels
where the state evolves via a general stochastic process, not necessarily
Markov or ergodic. The states are assumed to be unknown to the transmitter and
the receiver, but the underlying probability distributions are known. For this
setup, we derive an upper bound on the feedback error exponent and the feedback
capacity with variable-length codes. The bounds are expressed in terms of the
directed mutual information and directed relative entropy. The bounds on the
error exponent are simplified to Burnashev's expression for discrete memoryless
channels. Our method relies on tools from the theory of martingales to analyze
a stochastic process defined based on the entropy of the message given the past
channel's outputs.
