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 --> infinity) 1/n D-alpha (p((n)) parallel to 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 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 alpha --> 1 and as alpha down arrow 0, Similarly, we provide a formula for the Renyi entropy rate lim(n --> infinity) 1/n H-alpha(p((n))) of Markov sources and examine its limits as alpha --> 1 and as alpha down arrow 0, Finally, we briefly provide an application to source coding.
renyi's divergence and entropy rates nonnegative matrices perron-frobenius theory kullback-leibler divergence rate shannon and hartley entropy rates time-invariant markov sources