Limit theorems for Hawkes processes including inhibition

TL;DR

This paper establishes limit theorems such as Law of Large Numbers, Central Limit Theorem, and large deviations for non-linear Hawkes processes that include both excitation and inhibition effects, extending previous results.

Contribution

It introduces a renewal-based approach to analyze Hawkes processes with signed kernels, covering inhibition effects not previously addressed in the literature.

Findings

Law of Large Numbers for signed Hawkes processes

Central Limit Theorem established for these processes

Explicit examples demonstrating the results

Abstract

In this paper we consider some non linear Hawkes processes with signed reproduction function (or memory kernel) thus exhibiting both self-excitation and inhibition. We provide a Law of Large Numbers, a Central Limit Theorem and large deviation results, as time growths to infinity. The proofs lie on a renewal structure for these processes introduced in Costa et al. (2020) which leads to a comparison with cumulative processes. Explicit computations are made on some examples. Similar results have been obtained in the literature for self-exciting Hawkes processes only.