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Temporal inhomogeneities in event sequences of natural and social phenomenahave been characterized in terms of interevent times and correlations betweeninterevent times. The inhomogeneities of interevent times have been extensivelystudied, while the correlations between interevent times, often calledcorrelated bursts, are far from being fully understood. For measuring thecorrelated bursts, two relevant approaches were suggested, i.e., memorycoefficient and burst size distribution. Here a burst size denotes the numberof events in a bursty train detected for a given time window. Empiricalanalyses have revealed that the larger memory coefficient tends to beassociated with the heavier tail of burst size distribution. In particular,empirical findings in human activities appear inconsistent, such that thememory coefficient is close to $0$, while burst size distributions follow apower law. In order to comprehend these observations, by assuming theconditional independence between consecutive interevent times, we derive theanalytical form of the memory coefficient as a function of parametersdescribing interevent time and burst size distributions. Our analytical resultcan explain the general tendency of the larger memory coefficient beingassociated with the heavier tail of burst size distribution. We also find thatthe apparently inconsistent observations in human activities are compatiblewith each other, indicating that the memory coefficient has limits to measurethe correlated bursts.

Limits of memory coefficient in measuring correlated bursts