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

TL;DR

This paper investigates the long-time behavior of multivariate Hawkes processes with power-law decaying long-range interactions, combining advanced mathematical techniques to understand their asymptotics, relevant for neural network modeling.

Contribution

It introduces a detailed analysis of Hawkes processes with long-range interactions, extending existing methods to account for power-law decay and long-term dynamics.

Findings

Derived asymptotic behavior for long-range Hawkes processes

Connected process properties to $ ext{alpha}$-stable laws

Applied Tauberian theorem to long-range interactions

Abstract

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 $1+\alpha.$ 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.