  Channels with noisy duplications have recently been used to model the
nanopore sequencer. This paper extends some foundational information-theoretic
results to this new scenario. We prove the asymptotic equipartition property
(AEP) for noisy duplication processes based on ergodic Markov processes. A
consequence is that the noisy duplication channel is information stable for
ergodic Markov sources, and therefore the channel capacity constrained to
Markov sources is the Markov-constrained Shannon capacity. We use the AEP to
estimate lower bounds on the capacity of the binary symmetric channel with
Bernoulli and geometric duplications using Monte Carlo simulations. In
addition, we relate the AEP for noisy duplication processes to the AEP for
hidden semi-Markov processes.
