Study of discrete-time Hawkes process and its compensator

TL;DR

This paper investigates the asymptotic properties and moment generating function of the discrete-time Hawkes process, providing bounds and convergence results to deepen understanding of its behavior.

Contribution

It offers new theoretical insights into the asymptotic behavior and MGF of the DTHP, extending knowledge from continuous-time to discrete-time models.

Findings

Derived bounds for the MGF of DTHP

Established convergence results for scaled logarithmic MGF

Analyzed the asymptotic behavior of DTHP and its compensator

Abstract

The discrete-time Hawkes process (DTHP) is a sub-class of $g$-functions that serves as a discrete-time version of the continuous-time Hawkes process (CTHP). Like the CTHP, the DTHP also has the self-exciting property and its intensity depends on the entire history. In this paper, we study the asymptotic behaviour of the DTHP and its compensator. We further analyse the moment generating function (MGF) of the DTHP and obtain some bounds and convergence results on the scaled logarithmic MGF of the DTHP.