A univariate Hawkes process is a simple point process that is self-excitingand has clustering effect. The intensity of this point process is given by thesum of a baseline intensity and another term that depends on the entire pasthistory 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 stationaryHawkes processes in the asymptotic regime where the baseline intensity islarge. The limit is a non-Markovian Gaussian process with dependent increments.We use the resulting approximation to study an infinite-server queue withhigh-volume Hawkes traffic. We show that the queue length process can beapproximated by a Gaussian process, for which we compute explicitly thecovariance function and the steady-state distribution. We also extend ourresults to multivariate stationary Hawkes processes and establish limittheorems for infinite-server queues with multivariate Hawkes traffic.
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