Distribution Related to All Samples and Extreme Events in the ETAS Cluster

The probabilistic approach of statistical seismology plays a fundamental role in ana-lyzing the evolution of an earthquake sequence at different scales to deliver reliableforecasts of complex earthquake phenomena. This study contributes to the field byextending previous results relative to theextreme events probability. We derive theexplicit formulation of the probability ofextreme events in any seismic cluster gen-erated by the epidemic-type aftershock sequence model, as a function of time, space,and magnitude. The results give insights into understanding the distinguishing fea-tures between mainshocks and foreshocks, thus helping to shed light on earthquake prediction.