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 quantity is the fixed-window hitting probability $H_T\left(u\right)$, the probability that a cluster born at time $-u$ contributes at least one event to a window of length $T$. For offspring waiting-time tails $\mathbb{P}(S>t)\sim t^{-\theta }$ and fertility tails $\mathbb{P}(\kappa >x)\sim x^{-\gamma }$, a finite-mean-lifetime argument rigorously establishes stationarity for $1<\gamma <2$ and $\theta >\gamma$, while the finite-memory, finite-variance critical regime is nonstationary. A direct asymptotic analysis of the exact fixed-window nonlinear renewal equation identifies the precise local-finiteness boundary $\theta =\gamma -1$ in the infinite-fertility-variance regime. Below this boundary, $H_T$ is nonintegrable. At the marginal point $\theta =\gamma -1$, a refined Laplace-Tauberian analysis yields $H_T\left(u\right)\sim \frac{{[lnu]}^{1/(\gamma -1)}}{u}$, which is also nonintegrable, demonstrating that the critical Hawkes process remains nonstationary at the boundary $\theta =\gamma -1$. Direct numerical solutions of the exact renewal equation confirm the predicted power-law and logarithmic regimes across the transition. Thus broad fertility fluctuations can sustain a locally finite stationary critical Hawkes process with positive immigration even though its stationary mean intensity is infinite.

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• Critical phenomena
• Geophysics
• Phase diagrams
• Scaling laws of complex systems