Long-time asymptotics for multivariate Hawkes processes with long-range interactions

Nadia Belmabrouk

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arXiv · 2603.05853

Long-time asymptotics for multivariate Hawkes processes with long-range interactions

Nadia Belmabrouk

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Abstract

We consider a system of interacting particles on an infinite graph, modeled by a multivariate Hawkes process with long-range interactions, where the interaction strength decays as a power law of the inter-particle distance with exponent $1+\alpha$. This model is more intricate and realistic for some applications, such as neural networks, where long-range connections are present. Our main focus is to characterize the long-time behavior of the system depending on the range of the interactions. These results correspond to laws of large numbers. We prove that long-range interactions affect the limiting behavior in the subcritical case but not in the supercritical case. The proofs of our results use properties of $\alpha$-stable laws, and Tauberian methods for Laplace transforms.

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Nadia Belmabrouk. 2026-03-06. Long-time asymptotics for multivariate Hawkes processes with long-range interactions. https://arxiv.org/abs/2603.05853

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