Jeffrey’s Divergence Between Fractionally Integrated White Noises

Abstract

Jeffrey’s divergence (JD) is used in many applications, from change detection to classification. Several studies were done on the JD between ergodic wide-sense stationary autoregressive and moving average (ARMA) processes. It was shown that the derivate of the JD between the probability density functions of k consecutive samples of two ARMA processes tends to the so-called asymptotic JD increment. This latter is enough to compare the processes and amounts to calculating the power of the first process filtered by the inverse filter associated with the second process and conversely. In this paper, our purpose is to study if this result can be extended to ARFIMA processes. As a first step, a special case, namely the JD between wide-sense stationary fractionally integrated white noises, is addressed. The influences of the process parameters on the asymptotic JD increment are analyzed. Our investigations validate the inverse filtering interpretation of the JD.