Large deviations for the mean-field limit of Hawkes processes

TL;DR

This paper establishes a large deviation principle for the mean-field limit of high-dimensional nonlinear Hawkes processes, which are point processes with history-dependent intensities, in the regime where the dimension tends to infinity.

Contribution

It introduces the first large deviation results for the mean-field limit of multidimensional nonlinear Hawkes processes as the dimension grows.

Findings

Large deviation principle derived for the mean-field limit.

Characterization of the asymptotic behavior of Hawkes processes in high dimensions.

Insights into the probability of rare events in complex point process systems.

Abstract

Hawkes processes are a class of simple point processes whose intensity depends on the past history, and is in general non-Markovian. Limit theorems for Hawkes processes in various asymptotic regimes have been studied in the literature. In this paper, we study a multidimensional nonlinear Hawkes process in the asymptotic regime when the dimension goes to infinity, whose mean-field limit is a time-inhomogeneous Poisson process, and our main result is a large deviation principle for the mean-field limit.