On Analytic Properties of Entropy Rate

Entropy rate is a real valued functional on the space of discrete random sources for which it exists. However, it lacks existence proofs and/or closed formulas even for classes of random sources which have intuitive parameterizations. A good way to overcome this problem is to examine its analytic properties relative to some reasonable topology. A canonical choice of a topology is that of the norm of total variation as it immediately arises with the idea of a discrete random source as a probability measure on sequence space. It is shown that both upper and lower entropy rate, hence entropy rate itself if it exists, are Lipschitzian relative to this topology, which, by well known facts, is close to differentiability. An application of this theorem leads to a simple and elementary proof of the existence of entropy rate of random sources with finite evolution dimension. This class of sources encompasses arbitrary hidden Markov sources and quantum random walks.