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arXiv · 2109.07126

Limit theorems for Hawkes processes including inhibition

Patrick Cattiaux·Laetitia Colombani·Manon Costa
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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.

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Patrick Cattiaux, Laetitia Colombani, Manon Costa. 2021-09-15. Limit theorems for Hawkes processes including inhibition. https://doi.org/10.1016/j.spa.2022.04.002

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