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

We study a multivariate Hawkes process with long-range interactions, where the interaction strength decays as a power-law in the distance of the particles with exponent $1+\alpha \mathrm{.}$ Our main focus is on the long-time asymptotic behavior of the system. The proofs of our results combine techniques developed for short-range interactions, properties of $\alpha$-stable laws, and a Tauberian theorem. This model is more intricate and realistic for some applications, such as neural networks, where long-range connections are present.