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

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 $1+α$. 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 $α$-stable laws, and Tauberian methods for Laplace transforms.