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A Hawkes Microfoundation for Multitype Inverse Gaussian Subordinators

Yingli Wang, Wei Xu, Lingjiong Zhu · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.10525 · cited by 0 works (OpenAlex) · open-access PDF

Korean

결론적으로, 우리는 결론적으로 많은 유형의 반대 가수 스토카스틱 시계를위한 유효한 이벤트 레벨 호크스 마이크로 기초를 제공합니다. 우리는 이벤트 계산 및 거의 중요한 다중 변수 선형 호크스 프로세스의 통합 강도가 마이크로 크로스 흥분에서 상속되어 발생할 수 있음을 보여줍니다. 우리는 재생 지연이 유효한 반대 가수 평균과 이민이 균형을 유지할 때이 프로세스가 다중 유형의 반대 가수 하위 계산자라고 부릅니다. 희귀 한 대규모 아토마 반대 가수 하위 계열의 다각적 계좌가 눈에 보인다. 독립적 인 단일 변수 좌표는 더 이상 특별한 경우로 상속됩니다. 우리는 그들의 특수 조건

Machine translation (Meta M2M100); the English original follows.

Abstract

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.

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Topics: Mathematics, Diagonal, Subordinator, Statistical physics, Univariate, Applied mathematics

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