Temporal inhomogeneities in event sequences of natural and social phenomena have been characterized in terms of interevent times and correlations between interevent times [1]. The inhomogeneities of interevent times have been extensively studied, while the correlations between interevent times, often called correlated bursts (CB), are far from being fully understood. Regarding the role of CB in temporal inhomogeneities, we numerically show that the strong CB, depicted by power-law burst size distributions, violates the well-established scaling relation between the power-law decaying autocorrelation function and the power-law interevent time distribution [2]. Next, it has been found that the empirical data sets for human activities show power-law burst size distributions but almost negligible memory coefficient. For understanding this, we derive an analytic form of the memory coefficient between consecutive interevent times as a function of parameters describing interevent time and burst size statistics, to conclude that the memory coefficient might have some limits in quantifying CB [3]. Finally, in order to completely characterize the event sequence, we develop a detection method of the hierarchical burst structure by exactly mapping the event sequence onto a rooted tree, implying no loss of information on the original event sequence [4]. We quantify the imbalance and time asymmetry of the tree and then estimate the kernel for merging bursts to model the event sequence from a novel perspective.
