Number of aftershocks in epidemic-type seismicity models

Let $V_M,(m_0)$ be the number of m>M aftershocks caused by $m_0$ event. We consider the $V_M,(m_0)$ 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 $M=m_0-\Delta$, we consider $M=m_a-\Delta$, where $m_a$ is the mode in the distribution of the strongest aftershock. Under these conditions we find the limit $V_M,(m_0)$ distribution at $m_0\gg1$. In the subcritical case, the limit distribution is extremely simple and identical to the $v(m_\Delta)$ distribution with a suitable magnitude $m=m_\Delta$. 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.