Convergence of Conditional Entropy for Long Range Dependent Markov Chains

TL;DR

This paper investigates how the convergence of conditional entropy to the entropy rate in long-range dependent Markov chains is slow, quantifies this rate, and links it to the chain's long-range dependence properties.

Contribution

It establishes the convergence rate of conditional entropy to the entropy rate in long-range dependent Markov chains and connects this rate to the chain's mixing time and long-range dependence.

Findings

Convergence rate is O(n^{2H-2}) where H is the Hurst parameter.

Mutual information between past and future is infinite if and only if the chain is long-range dependent.

Slow convergence of entropy measures is linked to the chain's long-range dependence.

Abstract

In this paper we consider the convergence of the conditional entropy to the entropy rate for Markov chains. Convergence of certain statistics of long range dependent processes, such as the sample mean, is slow. It has been shown in Carpio and Daley \cite{carpio2007long} that the convergence of the $n$-step transition probabilities to the stationary distribution is slow, without quantifying the convergence rate. We prove that the slow convergence also applies to convergence to an information-theoretic measure, the entropy rate, by showing that the convergence rate is equivalent to the convergence rate of the $n$-step transition probabilities to the stationary distribution, which is equivalent to the Markov chain mixing time problem. Then we quantify this convergence rate, and show that it is $O(n^{2H-2})$, where $n$ is the number of steps of the Markov chain and $H$ is the Hurst parameter. Finally, we show that due to this slow convergence, the mutual information between past and future is infinite if and only if the Markov chain is long range dependent. This is a discrete analogue of characterisations which have been shown for other long range dependent processes.