For finite-state indecomposable channels, Shannon's basic theorem, that transmission is possible at any rate less than channel capacity but not at any greater rate, is proved. A necessary and sufficient condition for indecomposability, from which it follows that every channel with finite memory is indecomposable, is given. An important tool is a modification, for some processes which are not quite stationary, of theorems of McMillan and Breiman on probabilities of long sequences in ergodic processes.
"Proof of Shannon's Transmission Theorem for Finite-State Indecomposable Channels." Ann. Math. Statist. 29 (4) 1209 - 1220, December, 1958. https://doi.org/10.1214/aoms/1177706452