Functional limit theorems for marked Hawkes point measures

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