Analysis of the autocorrelation function for time series with higher-order temporal correlations: An exponential case

Temporal correlations in the time series observed in various systems have been characterized by the autocorrelation function. Such correlations can be explained by interevent time distributions as well as by correlations between interevent times or higher-order temporal correlations. Despite its importance, the impact of higher-order temporal correlations on the autocorrelation function has been largely unexplored. For studying such impact, we focus on the bursts, i.e., clusters of rapidly occurring events within short time periods, and positive correlations between consecutive burst sizes. We devise a model generating a time series with correlated burst sizes by employing the copula method. We successfully derive the general analytical solution of the autocorrelation function of the model time series for arbitrary distributions of interevent times and burst sizes when consecutive burst sizes are correlated. For the demonstration of our analysis, we adopt exponential distributions of interevent times and burst sizes to find that the analytical solutions are in good agreement with numerical simulations. Our approach helps us to understand how higher-order temporal correlations affect the decaying behavior of the autocorrelation function.