Asymptotics for a Class of Self-Exciting Point Processes

TL;DR

This paper investigates the long-term behavior of a class of self-exciting point processes with nonlinear intensity, deriving fundamental asymptotic results such as laws of large numbers, CLT, and large deviations.

Contribution

It introduces a novel analysis of nonlinear self-exciting point processes, bridging stochastic dynamics and deterministic systems for asymptotic behavior.

Findings

Law of large numbers established for the process

Central limit theorem derived for fluctuations

Asymptotic tail probability estimates obtained

Abstract

In this paper, we study a class of self-exciting point processes. The intensity of the point process has a nonlinear dependence on the past history and time. When a new jump occurs, the intensity increases and we expect more jumps to come. Otherwise, the intensity decays. The model is a marriage between stochasticity and dynamical system. In the short-term, stochasticity plays a major role and in the long-term, dynamical system governs the limiting behavior of the system. We study the law of large numbers, central limit theorem, large deviations and asymptotics for the tail probabilities.