  Using the standard ETAS model of triggered seismicity, we present a rigorous
theoretical analysis of the main statistical properties of temporal clusters,
defined as the group of events triggered by a given main shock of fixed
magnitude m that occurred at the origin of time, at times larger than some
present time t. Using the technology of generating probability function (GPF),
we derive the explicit expressions for the GPF of the number of future
offsprings in a given temporal seismic cluster, defining, in particular, the
statistics of the cluster's duration and the cluster's offsprings maximal
magnitudes. We find the remarkable result that the magnitude difference between
the largest and second largest event in the future temporal cluster is
distributed according to the regular Gutenberg-Richer law that controls the
unconditional distribution of earthquake magnitudes. For earthquakes obeying
the Omori-Utsu law for the distribution of waiting times between triggering and
triggered events, we show that the distribution of the durations of temporal
clusters of events of magnitudes above some detection threshold \nu has a power
law tail that is fatter in the non-critical regime $n<1$ than in the critical
case n=1. This paradoxical behavior can be rationalised from the fact that
generations of all orders cascade very fast in the critical regime and
accelerate the temporal decay of the cluster dynamics.

Geophysics