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

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 applicati...

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Bibliographic Details
Published in:Queueing systems 2018-10, Vol.90 (1-2), p.161-206
Main Authors: Gao, Xuefeng, Zhu, Lingjiong
Format: Article
Language:English
Subjects:
Business and Management
Central limit theorem
Clustering
Computer Communication Networks
Control
Covariance
Gaussian process
Markov processes
Operations Research/Decision Theory
Probability Theory and Stochastic Processes
Queues
Queuing theory
Random variables
Seismology
Supply Chain Management
Systems Theory
Theorems
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Summary: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.
ISSN:0257-0130
1572-9443
DOI:10.1007/s11134-018-9570-5