Fluctuations and moderate deviations for the mean fields of Hawkes processes

TL;DR

This paper investigates the fluctuations and moderate deviations of the mean fields in Hawkes processes, providing insights into their probabilistic behavior beyond the mean-field limit.

Contribution

It offers a detailed analysis of the fluctuations and moderate deviations for the mean fields of Hawkes processes, extending understanding of their probabilistic properties.

Findings

Characterization of fluctuations around the mean-field limit.

Derivation of large deviation principles for moderate deviations.

Insights into the probabilistic behavior of Hawkes processes.

Abstract

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.