https://hal-univ-lemans.archives-ouvertes.fr/hal-02443841Horst, UlrichUlrichHorstInstitut für Mathematik [Berlin] - TU - Technische Universität BerlinXu, WeiWeiXuInstitut für Mathematik [Berlin] - TU - Technische Universität BerlinFunctional Limit Theorems for Marked Hawkes Point Measures *HAL CCSD2019Hawkes point measuremarked Hawkes processfunctional limit theorembudding microbes in a host[MATH] Mathematics [math][MATH.MATH-PR] Mathematics [math]/Probability [math.PR][QFIN.TR] Quantitative Finance [q-fin]/Trading and Market Microstructure [q-fin.TR][QFIN.GN] Quantitative Finance [q-fin]/General Finance [q-fin.GN]Xu, Wei2020-01-17 14:13:232020-12-28 10:22:042020-01-20 15:08:41enPreprints, Working Papers, ...application/pdf1This 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.