Renyi´s divergence and entropy rates for finite alphabet Markov sources

In this work, we examine the existence and the computation of the Renyi divergence rate, lim_n→∞ 1/n D_α (p⁽ⁿ⁾||q⁽ⁿ⁾), between two time-invariant finite-alphabet Markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p⁽ⁿ⁾ and q⁽ⁿ⁾, respectively. This yields a generalization of a result of Nemetz (1974) where he assumed that the initial probabilities under p⁽ⁿ⁾ and q⁽ⁿ⁾ are strictly positive. The main tools used to obtain the Renyi divergence rate are the theory of nonnegative matrices and Perron-Frobenius theory. We also provide numerical examples and investigate the limits of the Renyi divergence rate as α→1 and as α↓0. Similarly, we provide a formula for the Renyi entropy rate lim_n→∞ 1/n H _α(p⁽ⁿ⁾) of Markov sources and examine its limits as α→1 and as α↓0. Finally, we briefly provide an application to source coding