Quantification of limit theorems for Hawkes processes

2025/03/27 by Coutin, Laure, Massat, Benjamin, Réveillac, Anthony · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.21273

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 2-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 2-Wasserstein distance and analyze the asymptotic quality of this coupling in detail.

Cited by

• Local stability and rates of convergence to equilibrium for the Nonlinear Renewal Equation; applications to Hawkes processes

Related

• Some limit theorems for locally stationary Hawkes processes
• Large deviations for the mean-field limit of Hawkes processes
• Nonlinear Hawkes Processes
• Scaling limits for Hawkes processes and application to financial\n statistics
• A generalized central limit theorem for critical marked Hawkes processes