Analytically solvable autocorrelation function for correlated interevent times

Long-term temporal correlations observed in event sequences of natural and social phenomena have been characterized by algebraically decaying autocorrelation functions. Such temporal correlations can be understood not only by heterogeneous interevent times (IETs) but also by correlations between IETs. In contrast to the role of heterogeneous IETs on the autocorrelation function, yet little is known about the effects due to the correlations between IETs. In order to rigorously study these effects, we derive an analytic form of the autocorrelation function as a function of the memory coefficient between two consecutive IETs for an arbitrary form of the IET distribution, by adopting the Farlie-Gumbel-Morgenstern copula for the joint probability distribution of two consecutive IETs. Our analytic results are confirmed by numerical simulations for exponential and power-law IET distributions. For the power-law case, we find the tendency of the steeper decay of the autocorrelation function for the stronger correlation between IETs. Our analytic approach enables us to better understand long-term temporal correlations induced by the correlations between IETs.