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

Technical Summary: Critical Hawkes Processes with Random Fertilities

Problem Statement
The paper addresses a fundamental ambiguity in the theory of self-exciting Hawkes point processes at criticality (branching ratio $n=1$). While it is well-established that the stationary mean intensity diverges as $n \to 1$ (specifically $\bar{\lambda} = \nu/(1-n)$), leading to the common conclusion that critical Hawkes processes are non-stationary or ill-defined, this conclusion relies solely on the finiteness of the first moment. The paper investigates whether a critical Hawkes process can remain stationary in law (i.e., possess time-invariant distributions for finite-window counts) despite having an infinite mean activity. Specifically, it asks whether the superposition of clusters generated by an immigration process starting at $t=-\infty$ results in a locally finite point process, or if the infinite past contributes infinitely many events to any bounded observation window.

Methodology
The author employs the Poisson-cluster construction of Hawkes processes, where each immigrant generates a genealogical tree (cluster) of events. The analysis focuses on the fixed-window hitting probability, denoted $H_T(u)$, defined as the probability that a cluster born at time $-u$ contributes at least one event to a fixed observation window $[0, T]$.

The core technical approach involves:

1. Local Finiteness Criterion: Establishing that the whole-line construction defines a genuine point process if and only if $\int_0^\infty H_T(u) du < \infty$ for any finite $T$.
2. Asymptotic Analysis: Deriving the asymptotic behavior of $H_T(u)$ for large $u$ under specific tail assumptions for the memory kernel $\phi(t) \sim t^{-(1+\theta)}$ and the fertility distribution $P(\kappa > x) \sim x^{-\gamma}$.
3. Generating Function Formalism: Utilizing branching process theory and generating functions to relate the extinction probability of clusters to the memory kernel and fertility distribution. This involves solving integral equations (e.g., $R(t) - (\phi * R)(t) \approx a(t)$) to determine the decay rates of survival probabilities and hitting probabilities.
4. Regime Classification: Distinguishing between regimes where the cluster lifetime mean is finite, regimes where the hitting probability is comparable to the lifetime survival probability (finite-memory/finite-variance), and regimes requiring direct asymptotic estimation of $H_T(u)$.

Key Contributions and Results
The paper derives precise conditions for the stationarity in law of critical Hawkes processes with heavy-tailed fertilities and memory kernels. The results are summarized by the interplay between the fertility tail exponent $\gamma$ (where $P(\kappa > x) \sim x^{-\gamma}$) and the memory tail exponent $\theta$ (where $\phi(t) \sim t^{-(1+\theta)}$).

1. Sufficient Condition via Finite Mean Lifetime:
   The paper proves that if the expected cluster lifetime $E[L]$ is finite, the process is stationary in law. This occurs when $1 < \gamma < 2$ and $\theta > \gamma$. In this regime, the condition $\int H_T(u) du < \infty$ is satisfied because $H_T(u)$ is bounded by the cluster survival probability $Q(u)$, which decays fast enough ($Q(u) \sim u^{-p}$ with $p>1$).

2. Non-Stationarity in Finite-Memory/Finite-Variance Regimes:
   For regimes with finite fertility variance ($\gamma \ge 2$) and finite memory ($\theta \ge 2$), the paper proves non-stationarity. Here, the hitting probability $H_T(u)$ is asymptotically comparable to the cluster survival probability $Q(u) \sim u^{-1}$. Since $\int u^{-1} du$ diverges, the local finiteness condition fails, and the process is not stationary in law.

3. Stabilization by Broad Fertility Fluctuations (Infinite Variance):
   The most significant finding is that broad fertility fluctuations (specifically $1 < \gamma < 2$) can stabilize critical dynamics in law even when the mean lifetime is infinite.

   • By directly analyzing $H_T(u)$ rather than relying on the lifetime tail, the paper establishes a sharper threshold for stationarity: $\theta > \gamma - 1$.
   • In the intermediate regime where $\gamma - 1 < \theta \le \gamma$, the mean cluster lifetime is infinite ($E[L] = \infty$), yet the fixed-window hitting probability decays sufficiently fast ($H_T(u) \sim u^{-(1+\theta)/\gamma}$) to ensure $\int H_T(u) du < \infty$.
   • Consequently, the process remains locally finite and stationary in law, producing sample paths with invariant finite-window distributions despite having an infinite mean intensity.

4. Classification Summary:

   • Stationary in Law (Proved): $1 < \gamma < 2$ and $\theta > \gamma$.
   • Stationary in Law (Heuristic/Asymptotic): $1 < \gamma < 2$ and $\gamma - 1 < \theta \le \gamma$.
   • Non-Stationary (Proved): $\gamma \ge 2$ and $\theta \ge 2$.
   • Non-Stationary (Heuristic): $\gamma > 2$ with $0 < \theta < 2$, and $1 < \gamma < 2$ with $0 < \theta \le \gamma - 1$.

Significance and Claims
The paper claims to resolve the logical gap between the divergence of the mean intensity and the existence of the process itself. It demonstrates that the standard conclusion—that critical Hawkes processes are non-stationary—is not generally valid. Instead, stationarity in law is controlled by the local finiteness of the infinite-past Poisson-cluster construction, not by the finiteness of the mean intensity.

The author argues that broad fertility fluctuations can "stabilize" critical Hawkes dynamics. This implies that empirical observations of superlinear fluctuations, divergent sample averages, or large endogenous cascades in systems like seismicity (ETAS models), financial markets, or neuronal avalanches do not necessarily indicate non-stationarity of the underlying law. Rather, these signatures may be the observable manifestations of a stationary heavy-tailed critical regime. The paper provides the mathematical criteria ($\theta > \gamma - 1$) to distinguish between true non-stationarity and stationary criticality with infinite mean, offering a refined framework for interpreting near-critical calibrations in natural and social systems.