Quantification of limit theorems for Hawkes processes
Abstract
In this article, we fill a gap in the literature regarding quantitative functional central limit theorems (qfCLT) for Hawkes processes by providing an upper bound for the convergence of a nearly unstable Hawkes process toward a Cox-Ingersoll-Ross (CIR) process. Note that in this case no speed of convergence has been established even for one-dimensional marginals; we provide in this paper a control in terms of a supremum norm in -Wasserstein distance. To do so, we make use of the so-called Poisson imbedding representation and provide a qfCLT formulation in terms of a Brownian sheet. Incidentally, we construct an optimal coupling between a rescaled bi-dimensional Poisson random measure and a Brownian sheet with respect to the -Wasserstein distance and analyze the asymptotic quality of this coupling in detail.
Cite
@article{arxiv.2503.21273,
title = {Quantification of limit theorems for Hawkes processes},
author = {Laure Coutin and Benjamin Massat and Anthony Réveillac},
journal= {arXiv preprint arXiv:2503.21273},
year = {2025}
}