Fertility Heterogeneity as a Mechanism for Power Law Distributions of Recurrence Times

Abstract: We study the statistical properties of recurrence times in the self-excited Hawkes conditional Poisson process, the simplest extension of the Poisson process that takes into account how the past events influence the occurrence of future events. Specifically, we analyze the impact of the power law distribution of fertilities with exponent \alpha, where the fertility of an event is the number of aftershocks of first generation that it triggers, on the probability distribution function (pdf) f(\tau) of the recurrence times \tau between successive events. The other input of the model is an exponential Omori law quantifying the pdf of waiting times between an event and its first generation aftershocks, whose characteristic time scale is taken as our time unit. At short time scales, we discover two intermediate power law asymptotics, f(\tau) ~ \tau^{-(2-\alpha)} for \tau << \tau_c and f(\tau) ~ \tau^{-\alpha} for \tau_c << \tau << 1, where \tau_c is associated with the self-excited cascades of aftershocks. For 1 << \tau << 1/\nu, we find a constant plateau f(\tau) ~ const, while at long times, 1/\nu < \tau, f(\tau) ~ e^{-\nu \tau} has an exponential tail controlled by the arrival rate \nu of exogenous events. These results demonstrate a novel mechanism for the generation of power laws in the distribution of recurrence times, which results from a power law distribution of fertilities in the presence of self-excitation and cascades of triggering.