Recursive computation of the Hawkes cumulants

Closed-form modeling of neuronal spike train statistics using multivariate Hawkes cumulants

We derive exact analytical expressions for the cumulants of any orders of neuronal membrane potentials driven by spike trains in a multivariate Hawkes process model with excitation and inhibition. Such expressions can be used for the prediction and sensitivity analysis of the statistical behavior of the model over time, and to estimate the probability densities of neuronal membrane potentials using Gram-Charlier expansions. Our results are shown to provide a better alternative to Monte Carlo estimates via stochastic simulations, and computer codes based on combinatorial recursions are included…

An algorithm for the computation of joint Hawkes moments with exponential kernel

The purpose of this paper is to present a recursive algorithm and its implementation in Maple and Mathematica for the computation of joint moments and cumulants of Hawkes processes with exponential kernels. Numerical results and computation times are also discussed. Obtaining closed form expressions can be computationally intensive, as joint fifth cumulant and moment formulas can be respectively expanded into up to 3,288 and 27,116 summands.

可交换对Hawkes参考模型与硬球碰撞统计的累积量障碍

An Exchangeable Pair-Hawkes Reference Model and a Cumulant Obstruction for Hard-Sphere Collision Statistics

本文提出以无序粒子对为索引的可交换Hawkes参考模型，推导其相关性质，发现硬球碰撞统计的Sonine基准无法被该模型重现，通过数值模拟验证了相关结论。

复合多元Hawkes过程：大偏差与稀有事件模拟

Compound Multivariate Hawkes Processes: Large Deviations and Rare Event Simulation

本文为随机标记多元Hawkes过程诱导的多元复合过程建立大偏差原理，证明极限累积量的本质光滑性，并应用其推导多元风险过程的边际破产概率渐近结果，同时提出基于重要性抽样的稀有事件模拟方法并证明其渐近有效性。

Normal approximation of compound Hawkes functionals

We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and non-negative integrands, our estimates involve only the third moment of integrand in addition to a variance term using a square norm of the integrand. As a consequence, we are able to observe a "third moment phenomenon" in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the curren…

高斯近似与泊松脉冲噪声的中等偏差及其在复合广义Hawkes过程中的应用

Gaussian Approximation and Moderate Deviations of Poisson Shot Noises with Application to Compound Generalized Hawkes Processes

本文研究了泊松脉冲噪声的高斯近似与中等偏差，推导了相关不等式，并应用于广义复合Hawkes过程。