Smooth Rényi entropy of ergodic quantum information sources

We investigate the recently introduced notion of smooth Rényi entropy for the case of ergodic information sources, thereby generalizing previous work which concentrated mainly on i.i.d. information sources. We will actually consider ergodic quantum information sources, of which ergodic classical information sources are a special case. We prove that the average smooth Rényi entropy rate will approach the entropy rate of a stationary, ergodic source, which is equal to the Shannon entropy rate for a classical source and the von Neumann entropy rate for a quantum source