Functional Central Limit Theorems for Stationary Hawkes Processes and Application to Infinite-Server Queues
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AbstractA univariate Hawkes process is a simple point process that is self-exciting and has a 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 processes have 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.
All Author(s) ListGao X., Zhu L.
Journal nameQueueing Systems
Year2018
Month10
Volume Number90
Issue Number1-2
PublisherSpringer
Pages161 - 206
ISSN0257-0130
eISSN1572-9443
LanguagesEnglish-United Kingdom
KeywordsGaussian limits, Functional central limit theorem, Stationary Hawkes processes, Infinite-server queues