# Some asymptotic results for nonlinear Hawkes processes

## TL;DR

This paper investigates the asymptotic behavior of nonlinear Hawkes processes under a regime of large baseline intensity and small excitation, providing new insights into their fluctuations and deviations.

## Contribution

It introduces a novel asymptotic regime for nonlinear Hawkes processes, analyzing fluctuations, large deviations, and moderate deviations in this context.

## Key findings

- Derived asymptotic results for nonlinear Hawkes processes
- Characterized fluctuations and deviations in the large intensity regime
- Applicable to large networks and mean process analysis

## Abstract

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.