Fluctuations and moderate deviations for the mean fields of Hawkes processes

The Hawkes process is a counting process that has self- and mutually-exciting features with many applications in various fields. In recent years, there have been many interests in the mean-field results of the Hawkes process and its extensions. It is known that the mean-field limit of a multivariate nonlinear Hawkes process is a time-inhomogeneous Poisson process. In this paper, we study the fluctuations for the mean fields and the large deviations associated with the fluctuations, i.e., the moderate deviations.