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Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

Xuefeng Gao () and Lingjiong Zhu ()
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Xuefeng Gao: The Chinese University of Hong Kong
Lingjiong Zhu: Florida State University

Queueing Systems: Theory and Applications, 2018, vol. 90, issue 1, No 6, 206 pages

Abstract: Abstract A 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.

Keywords: Stationary Hawkes processes; Functional central limit theorem; Infinite-server queues; Gaussian limits; 60F17; 60K25; 90B22 (search for similar items in EconPapers)
Date: 2018
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