An informatic approach to a long memory stationary process

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 Ip−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 Ip−f is infinite. For a stationary process with finite block entropy, Ip−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 Ip−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 Ip−f is that it is invariant under one-to-one transformation; this enables us to know the Ip−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 Ip−f of fractional Gaussian noise is infinite if and only if the Hurst parameter is H ∈ (1/2, 1).