标题： 长程依赖马尔可夫链的条件熵收敛性 显示英文标题

标题： Convergence of Conditional Entropy for Long Range Dependent Markov Chains

摘要： 在本文中，我们考虑马尔可夫链的条件熵向熵率收敛的问题。 长程依赖过程的某些统计量（如样本均值）的收敛速度较慢。 Carpio 和 Daley\cite{carpio2007long}证明了$n$步转移概率向平稳分布收敛的速度较慢，但没有量化收敛速率。 我们通过证明收敛速率等价于$n$步转移概率向平稳分布的收敛速率，从而证明这种缓慢收敛也适用于信息论中的度量——熵率，而该速率又等价于马尔可夫链混合时间问题。 然后我们量化了这一收敛速率，并证明其为$O(n^{2H-2})$，其中$n$是马尔可夫链的步数，$H$是 Hurst 参数。 最后，我们证明由于这种缓慢收敛，过去和未来之间的互信息为无穷大当且仅当马尔可夫链是长程依赖的。 这是对其他长程依赖过程已有的表征的一个离散类比。 显示英文摘要

摘要： 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.