Long-time asymptotics for multivariate Hawkes processes with 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 $1+α.$ 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 $α$-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.

Theorems & Definitions (17)

• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['thm:sup2']}