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This paper establishes a functional law of large numbers and a functionalcentral limit theorem for marked Hawkes point measures and their correspondingshot noise processes. We prove that the normalized random measure can beapproximated in distribution by the sum of a Gaussian white noise process plusan appropriate lifting map of a correlated one-dimensional Brownian motion. TheBrownian results from the self-exiting arrivals of events. We apply our limittheorems for Hawkes point measures to analyze the population dynamics ofbudding microbes in a host.

Functional Limit Theorems for Marked Hawkes Point Measures