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The purpose of this paper is first to derive the expressions of the Chernoff, Bhattacharyya, Renyi and Sharma-Mittal divergences when comparing two probability density functions of vectors storing k consecutive samples of Gaussian ARMA processes. This can be useful for process comparison or statistical change detection issue. Then, we analyze the behaviors of the divergences when k increases and tends to infinity by using some results related to ARMA processes such as the Yule-Walker equations. Comments and illustrations are given.

In this paper, we propose to analyze the properties of the cross entropy between the probability density functions of k consecutive samples of wide-sense stationary Gaussian ARMA processes. It is shown that, when k increases, the cross entropy has a transient behavior before tending to an affine function of k whose slope is defined by the so-called asymptotic increment. The latter depends on the parameters of the processes and can be a reasonable choice to characterize the cross entropy. Some illustrations are then given.

Jeffrey’s divergence (JD), the symmetric Kullback-Leibler (KL) divergence, has been used in a wide range of applications. In recent works, it was shown that the JD between probability density functions of k successive samples of autoregressive (AR) and/or moving average (MA) processes can tend to a stationary regime when the number k of variates increases. The asymptotic JD increment, which is the difference between two JDs computed for k and \(k-1\) successive variates tending to a finite constant value when k increases, can hence be useful to compare the random processes. However, interpreting the value of the asymptotic JD increment is not an easy task as it depends on too many parameters, i.e. the AR/MA parameters and the driving-process variances. In this paper, we propose to compute the asymptotic JD increment between the processes that have been normalized so that their powers are equal to 1. Analyzing the resulting JD on the one hand and the ratio between the original signal powers on the other hand makes the interpretation easier. Examples are provided to illustrate the relevance of this way to operate with the JD.