Number of aftershocks in epidemic-type seismicity models

Let V(M, m0) be the number of m>M aftershocks caused by m0 event. We consider the V(M, m0) distribution within epidemic-type seismicity models, ETAS(F). These models include the Gutenberg-Richter law for magnitude and Utsu law for average productivity of m0, but differ in the type of F distribution for the number v(m0) of direct aftershocks. The class of F is quite broad and includes both the Poisson distribution, which is the basis for the regular ETAS model, and its possible alternative, the Geometric distribution. Instead of the traditional threshold M=m0-d we consider M=ma-d, where ma is the mode in the distribution of the strongest aftershock. Under these conditions we find the limit V(M, m0) distribution at m0>>1. In the subcritical case, the limit distribution is extremely simple and identical to the v(m) distribution with a suitable magnitude m=m(d). Theoretical results of this kind are lacking even for the regular ETAS model. Our results provide an additional opportunity to test the type of F-distribution, Bath law, and the very concept of epidemic-type clustering.