Jensen-autocorrelation function for weakly stationary processes and applications

The Jensen-variance (JV) information based on Jensen's inequality and variance has been previously proposed to measure the distance between two random variables. Based on the relationship between JV distance and autocorrelation function of two weakly stationary process, the Jensen-autocovariance and Jensen-autocorrelation functions are proposed in this paper. Furthermore, the distance between two different weakly stationary processes is measured by the Jensen-cross-correlation function. Moreover, autocorrelation function is also considered for ARMA and ARFIMA processes, deriving explicit formulas for Jensen-autocorrelation function that only depends on model parametric space and lag, whose were also illustrated by numeric results. In order to study the usefulness of proposed functions, two real-life applications were considered: the Tree Ring and Southern Humboldt current ecosystem time series.