  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 white 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.
