数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (6): 2629-2648.doi: 10.1007/s10473-023-0619-0

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AN INFORMATIC APPROACH TO A LONG MEMORY STATIONARY PROCESS*

Yiming DING1,2, Liang WU3, Xuyan XIANG4,†   

  1. 1. College of Science, Wuhan University of Science and Technology, Wuhan 430081, China;
    2. Hubei Province Key Laboratory of System Science in Metallurgical Process, Wuhan University of Science and Technology, Wuhan 430065, China;
    3. Center of Statistical Research, School of Statistics, Southwestern University of Finance and Economics, Chengdu 611130, China;
    4. Hunan Province Cooperative Innovation Center for TCDDLEEZ, School of Mathematics and Physics Science, Hunan University of Arts and Science, Changde 415000, China
  • 收稿日期:2022-04-26 修回日期:2023-05-25 发布日期:2023-12-08
  • 通讯作者: †Xuyan XIANG, E-mail: xyxiang2001@126.com
  • 作者简介:Yiming DING, E-mail: dingym@wust.edu.cn; Liang WU, E-mail: wuliang@swufe.edu.cn
  • 基金资助:
    This work was supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry, the Key Scientific Research Project of Hunan Provincial Education Department (19A342), the National Natural Science Foundation of China (11671132, 61903309 and 12271418), the National Key Research and Development Program of China (2020YFA0714200), Sichuan Science and Technology Program (2023NSFSC1355), and the Applied Economics of Hunan Province. All authors are co-first authors of the article.

AN INFORMATIC APPROACH TO A LONG MEMORY STATIONARY PROCESS*

Yiming DING1,2, Liang WU3, Xuyan XIANG4,†   

  1. 1. College of Science, Wuhan University of Science and Technology, Wuhan 430081, China;
    2. Hubei Province Key Laboratory of System Science in Metallurgical Process, Wuhan University of Science and Technology, Wuhan 430065, China;
    3. Center of Statistical Research, School of Statistics, Southwestern University of Finance and Economics, Chengdu 611130, China;
    4. Hunan Province Cooperative Innovation Center for TCDDLEEZ, School of Mathematics and Physics Science, Hunan University of Arts and Science, Changde 415000, China
  • Received:2022-04-26 Revised:2023-05-25 Published:2023-12-08
  • Contact: †Xuyan XIANG, E-mail: xyxiang2001@126.com
  • About author:Yiming DING, E-mail: dingym@wust.edu.cn; Liang WU, E-mail: wuliang@swufe.edu.cn
  • Supported by:
    This work was supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry, the Key Scientific Research Project of Hunan Provincial Education Department (19A342), the National Natural Science Foundation of China (11671132, 61903309 and 12271418), the National Key Research and Development Program of China (2020YFA0714200), Sichuan Science and Technology Program (2023NSFSC1355), and the Applied Economics of Hunan Province. All authors are co-first authors of the article.

摘要: Long memory is an important phenomenon that arises sometimes in the analysis of time series or spatial data. Most of the definitions concerning the long memory of a stationary process are based on the second-order properties of the process. The mutual information between the past and future $I_{p-f}$ of a stationary process represents the information stored in the history of the process which can be used to predict the future. We suggest that a stationary process can be referred to as long memory if its $I_{p-f}$ is infinite. For a stationary process with finite block entropy, $I_{p-f}$ is equal to the excess entropy, which is the summation of redundancies that relate the convergence rate of the conditional (differential) entropy to the entropy rate. Since the definitions of the $I_{p-f}$ and the excess entropy of a stationary process require a very weak moment condition on the distribution of the process, it can be applied to processes whose distributions are without a bounded second moment. A significant property of $I_{p-f}$ is that it is invariant under one-to-one transformation; this enables us to know the $I_{p-f}$ of a stationary process from other processes. For a stationary Gaussian process, the long memory in the sense of mutual information is more strict than that in the sense of covariance. We demonstrate that the $I_{p-f}$ of fractional Gaussian noise is infinite if and only if the Hurst parameter is $H \in (1/2, 1)$.

关键词: mutual information between past and future, long memory, stationary process, excess entropy, fractional Gaussian noise

Abstract: Long memory is an important phenomenon that arises sometimes in the analysis of time series or spatial data. Most of the definitions concerning the long memory of a stationary process are based on the second-order properties of the process. The mutual information between the past and future $I_{p-f}$ of a stationary process represents the information stored in the history of the process which can be used to predict the future. We suggest that a stationary process can be referred to as long memory if its $I_{p-f}$ is infinite. For a stationary process with finite block entropy, $I_{p-f}$ is equal to the excess entropy, which is the summation of redundancies that relate the convergence rate of the conditional (differential) entropy to the entropy rate. Since the definitions of the $I_{p-f}$ and the excess entropy of a stationary process require a very weak moment condition on the distribution of the process, it can be applied to processes whose distributions are without a bounded second moment. A significant property of $I_{p-f}$ is that it is invariant under one-to-one transformation; this enables us to know the $I_{p-f}$ of a stationary process from other processes. For a stationary Gaussian process, the long memory in the sense of mutual information is more strict than that in the sense of covariance. We demonstrate that the $I_{p-f}$ of fractional Gaussian noise is infinite if and only if the Hurst parameter is $H \in (1/2, 1)$.

Key words: mutual information between past and future, long memory, stationary process, excess entropy, fractional Gaussian noise

中图分类号: 

  • 60G10