A Full Space–Time–Magnitude Extreme-Event Distribution for an ETAS Cluster

Earthquake forecasting remains a fundamental challenge in statistical physics and geoscience due to the stochastic, clustered, and heavy-tailed nature of seismicity. A key limitation in current models is the lack of a rigorous characterization of extreme-event statistics in space–time–magnitude domains. In this Letter, we derive analytically the distribution of the largest event within a seismic cluster generated by a subcritical epidemic-type aftershock sequence model. Using a Neumann series expansion and a constructive bounding technique, we prove that the probability density η(t,x,y,m) decays as a power law in both time and space, η∼K(m)t−pr−ξ, under general heavy-tailed kernels with p > 1, ξ>2, and subcriticality condition |B(m)|<1. This result reveals the asymptotic structure of extreme seismic fluctuations and quantifies the decay rate of the most destructive events. Numerical simulations confirm our theory, providing a first-principles foundation for forecasting extreme events in earthquake-prone regions. Our findings have direct implications for probabilistic hazard assessment and represent a critical step toward predictive seismology grounded in statistical mechanics.