LIMIT THEOREMS FOR HAWKES PROCESSES INCLUDING INHIBITION

Patrick Cattiaux (1) , Laetitia Colombani (1) , Manon Costa (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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-Graham-Marsalle-Tran (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.

Mots clés: Hawkes processes; inhibition; renewal theory; limit theorems.

Domaines: Probabilités [math.PR]

Patrick Cattiaux, Laetitia Colombani, Manon Costa. LIMIT THEOREMS FOR HAWKES PROCESSES INCLUDING INHIBITION. 2021. ⟨hal-03344416⟩