Asymptotic Analysis of Discrete-Time Hawkes Process

In a discrete-time setting, we consider an arrival process $\left\{{\xi }_n\right|n=1,2,\dots \}$, which models the occurrence of events, and a corresponding point process $\left\{H_n\right|n=1,2,\dots \}$, known as the discrete-time Hawkes process. These two stochastic processes are related by $H_n={\sum }_{i=1}^n{\xi }_i$, and exhibit a self-exciting property. In particular, we study the limiting behavior of the arrival process and establish the Large Deviation Principle for the discrete-time Hawkes process. We also illustrate an application in which insurance claims are modeled using the discrete-time Hawkes process and analyze its behavior.