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Functional Limit Theorems for Marked Hawkes Point Measures

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  • Ulrich Horst

    (Institut für Mathematik [Berlin] - TU - Technical University of Berlin / Technische Universität Berlin)

  • Wei Xu

    (Institut für Mathematik [Berlin] - TU - Technical University of Berlin / Technische Universität Berlin)

Abstract

This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian wihte noise process plus an appropriate lifting map of a correlated one-dimensional Brownian motion. The Brownian results from the self-exiting arrivals of events. We apply our limit theorems for Hawkes point measures to analyze the population dynamics of budding microbes in a host.

Suggested Citation

  • Ulrich Horst & Wei Xu, 2019. "Functional Limit Theorems for Marked Hawkes Point Measures ," Working Papers hal-02443841, HAL.
    • Handle: RePEc:hal:wpaper:hal-02443841
    Note: View the original document on HAL open archive server: https://univ-lemans.hal.science/hal-02443841
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    References listed on IDEAS

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    Cited by:

    1. Ulrich Horst & Wei Xu, 2019. "The Microstructure of Stochastic Volatility Models with Self-Exciting Jump Dynamics," Papers 1911.12969, arXiv.org.

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