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.

翻译：我们研究自激负二项过程的连续时间极限，并探讨其强度分布的临界性质。在此极限下，该过程转化为标记霍克斯过程。标记的概率质量函数包含参数$\omega$，当$\omega\to 0$时，该过程退化为“纯”霍克斯过程。我们研究了标记霍克斯过程在拉普拉斯表示下主方程的拉格朗日-沙皮特方程在其临界点附近的性质，并将先前关于中间渐近区域强度概率密度函数幂律标度的发现推广至记忆核为任意有限个指数函数叠加的情形。基于时间重标度定理，我们开发了一种针对标记霍克斯过程的高效采样方法，并对幂律指数进行了验证。