发表机构
Fudan University; Beijing Institute of Technology; Florida State University（复旦大学; 北京理工大学; 佛罗里达州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为多类型逆高斯子序提供事件级Hawkes微观基础，证明近临界多变量Hawkes过程收敛到该子序，并揭示其机制与标度理论关系。
AI 中文摘要
我们为多类型逆高斯随机时钟提供了一个事件层面的Hawkes微观基础。我们证明了当繁殖延迟具有有限均值且移民被平衡使得罕见的大簇仍然可见时，近临界的多变量线性Hawkes过程的事件计数和积分强度联合收敛到一个多变量纯跳跃子序。极限坐标之间的依赖性继承自微观交叉激发，且该极限具有布朗加性场首达时表示。我们将此过程称为多类型逆高斯子序。其对角情形由独立的经典逆高斯子序组成，单变量模型作为其进一步特例。平方根特化恢复了Abi Jaber--Attal--Rosenbaum（《应用概率年鉴》第36卷第4期，第3635--3660页，2026年）在超粗糙平方根模型边界处获得的逆高斯时钟。我们的簇证明揭示了潜在机制：有限方差的近临界分支产生罕见但宏观的簇，而它们的内部时序在观测尺度上消失，因此每个簇变成一个跳跃。我们还阐明了与Xu（arXiv:2412.14459）的多变量Hawkes标度理论的关系：其对角原子条件产生对角特化，而真正耦合的极限需要单独的唯一性论证。此外，在共同的倾斜稳定性条件下，我们将该理论中的高强度假设替换为较弱的累积活动条件，并提供了当势具有原子时所需的Riccati识别。我们加强了计数和补偿器的收敛性，并将标量结论扩展到有限方差的年龄依赖分支簇。
英文摘要
We provide an event-level Hawkes microfoundation for a multitype inverse-Gaussian stochastic clock. We show that the event counts and integrated intensities of nearly critical multivariate linear Hawkes processes converge jointly to a multivariate pure-jump subordinator when reproduction delays have finite mean and immigration is balanced so that rare large families remain visible. The dependence among the limiting coordinates is inherited from microscopic cross-excitation, and the limit admits a Brownian additive-field first-passage representation. We call this process the multitype inverse-Gaussian subordinator. Its diagonal case consists of independent classical inverse-Gaussian subordinators, with the univariate model as a further special case. The square-root specialization recovers the inverse-Gaussian clock obtained at the boundary of hyper-rough square-root models by Abi Jaber--Attal--Rosenbaum (\textit{Ann. Appl. Probab.} \textbf{36}(4): 3635--3660, 2026). Our cluster proof exposes the underlying mechanism: finite-variance near-critical branching creates rare but macroscopic families, while their internal timing disappears on the observation scale, so each family becomes a jump. We also clarify the relation with the multivariate Hawkes scaling theory of Xu (arXiv:2412.14459): its diagonal-atom condition yields the diagonal specialization, whereas the genuinely coupled limit requires a separate uniqueness argument. In addition, under a common tilted-stability condition, we replace the high-intensity assumption in that theory by a weaker accumulated-activity condition and provide the Riccati identification needed when the potential has an atom. We strengthen convergence of the count and compensator and extend the scalar conclusion to finite-variance age-dependent branching clusters.
Comments35 pages
