A quantity of interest to characterise continuous-valued stochastic processes is the differential entropy rate. The rate of convergence of many properties of LRD processes is slower than might be expected, based on the intuition for conventional processes, e.g. Markov processes. Is this also true of the entropy rate?
In this paper we consider the properties of the differential entropy rate of stochastic processes that have an autocorrelation function that decays as a power law. We show that power law decaying processes with similar autocorrelation and spectral density functions, Fractional Gaussian Noise and ARFIMA(0,d,0), have different entropic properties, particularly for negatively correlated parameterisations. Then we provide an equivalence between the mutual information between past and future and the differential excess entropy for stationary Gaussian processes, showing the finiteness of this quantity is the boundary between long and short range dependence. Finally, we analyse the convergence of the conditional entropy to the differential entropy rate and show that for short range dependence that the rate of convergence is of the order $O(n^{-1})$, but it is slower for long range dependent processes and depends on the Hurst parameter.
Network Coding Capacity Regions via Entropy Functions
在本文中，我们使用熵函数来描述一般网络编码问题的速率容量元组集，这些元组可以是零解码错误，也可以是零解码错误。我们表明，当源是同位的，由Yeung，A First Course in信息论，第15.5节（2002年）得到的外边界是紧的，零误差可实现和消失误差可实现的速率容量元组的集合是相同的。我们还讨论了一组零错误和消失错误可实现的速率容量元组的网络编码问题的线性编码约束，路由约束（其中一些或所有节点只能执行路由）和保密约束。最后，我们表明，即使是显然简单的网络，最佳代码的设计可能是困难的。特别是，我们证明，对于增量组播问题和单源安全网络编码问题，表征的可实现集是非常困难的，线性网络代码可能不是最佳的。
On the Estimation of Information Measures of Continuous Distributions
Institute of Telecommunications, TU Wien（维也纳工业大学电信研究所）；Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes（巴黎萨克雷大学、法国国家科学研究中心、中央理工-高等电力学院、信号与系统实验室）；Montreal Institute for Learning Algorithms (Mila), Université de Montréal（蒙特利尔学习算法研究所（Mila）、蒙特利尔大学）；Acoustics Research Institute, Austrian Academy of Sciences（奥地利科学院声学研究所）
