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We study the statistical properties of recurrence times in the self-excitedHawkes conditional Poisson process, the simplest extension of the Poissonprocess that takes into account how the past events influence the occurrence offuture events. Specifically, we analyze the impact of the power lawdistribution of fertilities with exponent \alpha, where the fertility of anevent is the number of aftershocks of first generation that it triggers, on theprobability distribution function (pdf) f(\tau) of the recurrence times \taubetween successive events. The other input of the model is an exponential Omorilaw quantifying the pdf of waiting times between an event and its firstgeneration aftershocks, whose characteristic time scale is taken as our timeunit. 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-excitedcascades of aftershocks. For 1 << \tau << 1/\nu, we find a constant plateauf(\tau) ~ const, while at long times, 1/\nu < \tau, f(\tau) ~ e^{-\nu \tau} hasan exponential tail controlled by the arrival rate \nu of exogenous events.These results demonstrate a novel mechanism for the generation of power laws inthe distribution of recurrence times, which results from a power lawdistribution of fertilities in the presence of self-excitation and cascades oftriggering.

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