  Temporal inhomogeneities in event sequences of natural and social phenomena
have been characterized in terms of interevent times and correlations between
interevent times. The inhomogeneities of interevent times have been extensively
studied, while the correlations between interevent times, often called
correlated bursts, are far from being fully understood. For measuring the
correlated bursts, two relevant approaches were suggested, i.e., memory
coefficient and burst size distribution. Here a burst size denotes the number
of events in a bursty train detected for a given time window. Empirical
analyses have revealed that the larger memory coefficient tends to be
associated with the heavier tail of burst size distribution. In particular,
empirical findings in human activities appear inconsistent, such that the
memory coefficient is close to $0$, while burst size distributions follow a
power law. In order to comprehend these observations, by assuming the
conditional independence between consecutive interevent times, we derive the
analytical form of the memory coefficient as a function of parameters
describing interevent time and burst size distributions. Our analytical result
can explain the general tendency of the larger memory coefficient being
associated with the heavier tail of burst size distribution. We also find that
the apparently inconsistent observations in human activities are compatible
with each other, indicating that the memory coefficient has limits to measure
the correlated bursts.
