The Kullback-Leibler divergence rate between Markov sources

In this work, we provide a computable expression for the Kullback-Leibler divergence rate lim_n→∞1/nD(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. We illustrate it numerically and examine its rate of convergence. The main tools used to obtain the Kullback-Leibler divergence rate and its rate of convergence are the theory of nonnegative matrices and Perron-Frobenius theory. Similarly, we provide a formula for the Shannon entropy rate lim_n→∞1/nH(p⁽ⁿ⁾) of Markov sources and examine its rate of convergence.