Rényi entropy and divergence for VARFIMA processes based on characteristic and impulse response functions

Rényi entropy based on characteristic function has been used as an information measure contained in wide-sense and real stationary vector autoregressive and moving average (VARMA) processes. These classes of processes have been extended by fractionally integrated VARMA (VARFIMA) ones, composed of a VARMA process, a vector of fractional differencing parameters, and independent and identically distributed multivariate normal random errors. Such processes have often been used to explicitly account for persistence to incorporate long-term correlations into multivariate data. The purpose of this paper is to extend Rényi entropy from VARMA to VARFIMA processes, addressing long-memory behavior of time series by adding a fractional differencing parameter. The characteristic function of the process can be derived directly from the asymptotic form of the impulse response function using the Wold representation. Then, assuming multivariate Gaussian white noise with known fractional differencing, autoregressive and moving average matrix parameters, the differential and Rényi entropies and Kullback–Leibler and Rényi divergences were obtained by evaluating the variance-covariance matrix identified with VARFIMA process distribution. The influences of the fractional differencing parameters on the Rényi entropy increment were analyzed, as were comparisons between VARFIMA processes using the Kullback–Leibler and Rényi divergences. Finally, numerical examples and an application to U.S. daily temperature time series are presented.