Normal approximation of compound Hawkes functionals

TL;DR

This paper provides quantitative bounds in Wasserstein distance for normal approximations of stochastic integrals with respect to Hawkes processes, highlighting a third moment phenomenon and extending results to compound Hawkes processes.

Contribution

It introduces new bounds involving third moments for Hawkes process integrals and applies these to compound Hawkes processes, improving upon existing literature.

Findings

Bounds involve third moments and variance of integrands.

Faster convergence rates when the first cumulant vanishes.

Improved estimates for large time in compound Hawkes processes.

Abstract

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 non-negative integrands, our estimates involve only the third moment of integrand in addition to a variance term using a square 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.