  A univariate Hawkes process is a simple point process that is self-exciting
and has clustering effect. The intensity of this point process is given by the
sum of a baseline intensity and another term that depends on the entire past
history of the point process. Hawkes process has wide applications in finance,
neuroscience, social networks, criminology, seismology, and many other fields.
In this paper, we prove a functional central limit theorem for stationary
Hawkes processes in the asymptotic regime where the baseline intensity is
large. The limit is a non-Markovian Gaussian process with dependent increments.
We use the resulting approximation to study an infinite-server queue with
high-volume Hawkes traffic. We show that the queue length process can be
approximated by a Gaussian process, for which we compute explicitly the
covariance function and the steady-state distribution. We also extend our
results to multivariate stationary Hawkes processes and establish limit
theorems for infinite-server queues with multivariate Hawkes traffic.
