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

Author

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  • Xuefeng Gao

    (The Chinese University of Hong Kong)

  • Lingjiong Zhu

    (Florida State University)

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.

Suggested Citation

  • Xuefeng Gao & Lingjiong Zhu, 2018. "Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues," Queueing Systems: Theory and Applications, Springer, vol. 90(1), pages 161-206, October.
    • Handle: RePEc:spr:queues:v:90:y:2018:i:1:d:10.1007_s11134-018-9570-5
    DOI: 10.1007/s11134-018-9570-5
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Xu Sun & Yunan Liu, 2021. "Staffing many‐server queues with autoregressive inputs," Naval Research Logistics (NRL), John Wiley & Sons, vol. 68(3), pages 312-326, April.
    2. Laurent Lesage & Madalina Deaconu & Antoine Lejay & Jorge Augusto Meira & Geoffrey Nichil & Radu State, 2022. "Hawkes processes framework with a Gamma density as excitation function: application to natural disasters for insurance," Post-Print hal-03040090, HAL.
    3. Laurent Lesage & Madalina Deaconu & Antoine Lejay & Jorge Augusto Meira & Geoffrey Nichil & Radu State, 2022. "Hawkes Processes Framework With a Gamma Density As Excitation Function: Application to Natural Disasters for Insurance," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2509-2537, December.
    4. Selvamuthu, Dharmaraja & Pandey, Shamiksha & Tardelli, Paola, 2023. "Limit Theorems for an extended inverse Hawkes process with general exciting functions," Statistics & Probability Letters, Elsevier, vol. 197(C).
    5. Andrew Daw & Jamol Pender, 2019. "On the distributions of infinite server queues with batch arrivals," Queueing Systems: Theory and Applications, Springer, vol. 91(3), pages 367-401, April.
    6. Onno Boxma & Michel Mandjes, 2021. "Shot-noise queueing models," Queueing Systems: Theory and Applications, Springer, vol. 99(1), pages 121-159, October.
    7. Seol, Youngsoo, 2019. "Limit theorems for an inverse Markovian Hawkes process," Statistics & Probability Letters, Elsevier, vol. 155(C), pages 1-1.
    8. Wang, Haixu, 2022. "Limit theorems for a discrete-time marked Hawkes process," Statistics & Probability Letters, Elsevier, vol. 184(C).
    9. Li, Bo & Pang, Guodong, 2022. "Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 285-339.
    10. Cattiaux, Patrick & Colombani, Laetitia & Costa, Manon, 2022. "Limit theorems for Hawkes processes including inhibition," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 404-426.
    11. Luca Mucciante & Alessio Sancetta, 2023. "Estimation of an Order Book Dependent Hawkes Process for Large Datasets," Papers 2307.09077, arXiv.org.
    12. Youngsoo Seol, 2023. "Large Deviations for Hawkes Processes with Randomized Baseline Intensity," Mathematics, MDPI, vol. 11(8), pages 1-10, April.
    13. Raviar Karim & Roger J. A. Laeven & Michel Mandjes, 2021. "Exact and Asymptotic Analysis of General Multivariate Hawkes Processes and Induced Population Processes," Papers 2106.03560, arXiv.org.
    14. Youngsoo Seol, 2022. "Non-Markovian Inverse Hawkes Processes," Mathematics, MDPI, vol. 10(9), pages 1-12, April.
    15. Laurent Lesage & Madalina Deaconu & Antoine Lejay & Jorge Augusto Meira & Geoffrey Nichil & Radu State, 2020. "Hawkes processes framework with a Gamma density as excitation function: application to natural disasters for insurance," Working Papers hal-03040090, HAL.

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