We characterize a Hawkes point process with kernel proportional to the probability density function of Mittag-Leffler random variables. This kernel decays as a power law with exponent $\beta +1 \in (1,2]$. Several analytical results can be proved, in particular for the expected intensity of the point process and for the expected number of events of the counting process. These analytical results are used to validate algorithms that numerically invert the Laplace transform of the expected intensity as well as Monte Carlo simulations of the process. Finally, Monte Carlo simulations are used to derive the full distribution of the number of events. The algorithms used for this paper are available at {\tt https://github.com/habyarimanacassien/Fractional-Hawkes}.

翻译：我们用与Mittag-Leffler随机变量的概率密度函数成正比的鹰角进程。 这个内核衰减为权力法,有Expentent $\beta +1 ein(1,2美元 ) 。 有几个分析结果可以证明, 特别是点进程的预期强度和计算过程的预期事件数量。 这些分析结果被用来验证从数字上将预期强度变换拉普尔的算法以及该进程的蒙特卡洛模拟。 最后, Monte Carlo 模拟被用来获取事件数量的完整分布。 本文所用的算法可在以下网站查阅: https://github.com/habyarimanascien/Fractional-Hawkes} 。