2021/10/28 by Andrew Feutrill, Matthew Roughan, Feutrill, Andrew +1
Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Methods and Inference #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2110.14881
openalex publication_date 2021/10/28 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In this paper we consider the convergence of the conditional entropy to the\nentropy rate for Markov chains. Convergence of certain statistics of long range\ndependent processes, such as the sample mean, is slow. It has been shown in\nCarpio and Daley citecarpio2007long that the convergence of the n-step\ntransition probabilities to the stationary distribution is slow, without\nquantifying the convergence rate. We prove that the slow convergence also\napplies to convergence to an information-theoretic measure, the entropy rate,\nby showing that the convergence rate is equivalent to the convergence rate of\nthe n-step transition probabilities to the stationary distribution, which is\nequivalent to the Markov chain mixing time problem. Then we quantify this\nconvergence rate, and show that it is O(n2H-2), where n is the number of\nsteps of the Markov chain and H is the Hurst parameter. Finally, we show that\ndue to this slow convergence, the mutual information between past and future is\ninfinite if and only if the Markov chain is long range dependent. This is a\ndiscrete analogue of characterisations which have been shown for other long\nrange dependent processes.\n