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