Normal Approximation of Compound Hawkes Functionals

• Published in: Journal of theoretical probability Vol. 37; no. 1; pp. 549 - 581
• Main Authors: Khabou, Mahmoud, Privault, Nicolas, Réveillac, Anthony
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2024; Springer Nature B.V; Springer
• Subjects: Approximation; Convergence; Estimates; Mathematical analysis; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; Random variables; Statistics; Normal approximation; 60G57; Stein’s method; 60G55; 60H07; Malliavin calculus; Hawkes processes; 60F05; normal approximation; Stein method; Malliavin calculus Mathematics Subject Classification : 60G55; Hawkes processes Stein method normal approximation Malliavin calculus Mathematics Subject Classification : 60G55
• ISSN: 0894-9840; 1572-9230
• DOI: 10.1007/s10959-022-01233-6

We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and nonnegative integrands, our estimates involve only the third moment of the integrand in addition to a variance term using a squared norm of the integrand. As a consequence, we are able to observe a “third moment phenomenon” in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time or have been obtained only for specific kernels such as the exponential or Erlang kernels.