Fractionally integrated process with power-law correlations in variables and magnitudes

Motivated by the fact that many empirical time series-including changes of heartbeat intervals, physical activity levels, intertrade times in finance, and river flux values-exhibit power-law anticorrelations in the variables and power-law correlations in their magnitudes, we propose a simple stochastic process that can account for both types of correlations. The process depends on only two parameters, where one controls the correlations in the variables and the other controls the correlations in their magnitudes. We apply the process to time series of heartbeat interval changes and air temperature changes and find that the statistical properties of the modeled time series are in agreement with those observed in the data.