Critical Hawkes Processes with Random Fertilities: Stationarity in Law Beyond Infinite Mean Activity

Didier Sornette

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arXiv · 2605.24595

Critical Hawkes Processes with Random Fertilities: Stationarity in Law Beyond Infinite Mean Activity

Didier Sornette

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Abstract

Genuinely critical dynamics have been proposed to organize many natural and social systems, yet exact criticality is usually thought to preclude stationarity because the mean activity diverges. I show that this conclusion is not generally valid for self-exciting Hawkes point processes. At criticality, stationarity in law is controlled not by the mean intensity, but by local finiteness of the infinite-past Poisson-cluster construction. The relevant object is the fixed-window hitting probability $H_T(u)$, the probability that a cluster born at time $-u$ contributes at least one event to a window of length $T$. For memory tails $\mathbb{P}(T>t)\sim t^{-\theta}$ and fertility tails $\mathbb{P}(\kappa>x)\sim x^{-\gamma}$, I prove stationarity for $1<\gamma<2$ and $\theta>\gamma$ via a finite-mean-lifetime criterion. In the finite-memory, finite-variance regime, $H_T(u)$ is asymptotically comparable to the cluster-survival probability, and the exact local-finiteness condition fails. A direct asymptotic analysis of $H_T$ gives the sharper condition $\theta>\gamma-1$ for stationarity to hold in the infinite-fertility-variance regime. Thus broad fertility fluctuations can stabilize critical Hawkes dynamics in law, producing locally finite stationary sample paths despite infinite mean activity.

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Didier Sornette. 2026-05-23. Critical Hawkes Processes with Random Fertilities: Stationarity in Law Beyond Infinite Mean Activity. https://arxiv.org/abs/2605.24595

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