Rnyi Entropy Rate of Stationary Ergodic Processes

In this paper, we examine the R & eacute;nyi entropy rate of stationary ergodic processes. For a special class of stationary ergodic processes, we prove that the R & eacute;nyi entropy rate always exists and can be approximated by its defining sequence at most polynomially; moreover, using the Markov approximation method, we show that the R & eacute;nyi entropy rate can be exponentially approximated by that of the Markov approximating sequence, as the Markov order goes to infinity. For the general case, by constructing a counterexample, we disprove the conjecture that the R & eacute;nyi entropy rate of a general stationary ergodic process always converges to its Shannon entropy rate as alpha goes to 1.