Self-exciting negative binomial distribution process and critical properties of intensity distribution

TL;DR

This paper analyzes the critical behavior of a self-exciting negative binomial process, showing its relation to marked Hawkes processes and revealing power-law scaling of intensity distributions near criticality.

Contribution

It introduces a continuous-time limit of the process, explores its critical properties, and develops an efficient sampling method for the marked Hawkes process.

Findings

Power-law scaling of intensity PDF near criticality

Extension to superpositions of exponential kernels

Efficient sampling method for marked Hawkes process

Abstract

We study the continuous time limit of a self-exciting negative binomial process and discuss the critical properties of its intensity distribution. In this limit, the process transforms into a marked Hawkes process. The probability mass function of the marks has a parameter $\omega$, and the process reduces to a "pure" Hawkes process in the limit $\omega\to 0$. We investigate the Lagrange--Charpit equations for the master equations of the marked Hawkes process in the Laplace representation close to its critical point and extend the previous findings on the power-law scaling of the probability density function (PDF) of intensities in the intermediate asymptotic regime to the case where the memory kernel is the superposition of an arbitrary finite number of exponentials. We develop an efficient sampling method for the marked Hawkes process based on the time-rescaling theorem and verify the power-law exponents.