  Hawkes process is a class of simple point processes with self-exciting and
clustering properties. Hawkes process has been widely applied in finance,
neuroscience, social networks, criminology, seismology, and many other fields.
In this paper, we study fluctuations, large deviations and moderate deviations
nonlinear Hawkes processes in a new asymptotic regime, the large intensity
function and the small exciting function regime. It corresponds to the large
baseline intensity asymptotics for the linear case, and can also be interpreted
as the asymptotics for the mean process of Hawkes processes on a large network.
