  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.
