We prove that the long-run behavior of Hawkes processes is fully determined by the average number and the dispersion of child events. For subcritical processes we provide FLLNs and FCLTs under minimal conditions on the kernel of the process with the precise form of the limit theorems depending strongly on the dispersion of child events. For a critical Hawkes process with weakly dispersed child events, functional central limit theorems do not hold. Instead, we prove that the rescaled intensity processes and rescaled Hawkes processes behave like CIR-processes without mean-reversion, respectively integrated CIR-processes. We provide the rate of convergence by establishing an upper bound on the Wasserstein distance between the distributions of rescaled Hawkes process and the corresponding limit process. By contrast, critical Hawkes process with heavily dispersed child events share many properties of subcritical ones. In particular, functional limit theorems hold. However, unlike subcritical processes critical ones with heavily dispersed child events display long-range dependencies.

翻译：本文证明霍克斯过程的长期行为完全由子事件的平均数和离散度决定。对于次临界过程，我们在关于过程核的最小条件下给出了泛函强大数定律和泛函中心极限定理，其极限定理的具体形式强烈依赖于子事件的离散度。对于子事件弱离散的临界霍克斯过程，泛函中心极限定理不成立。取而代之，我们证明重标度强度过程和重标度霍克斯过程分别表现为无均值回归的CIR过程与积分型CIR过程。通过建立重标度霍克斯过程分布与对应极限过程分布之间的Wasserstein距离上界，我们给出了收敛速率。相比之下，子事件强离散的临界霍克斯过程具有许多与次临界过程相似的性质，特别是泛函极限定理成立。然而与次临界过程不同，子事件强离散的临界过程表现出长程依赖性。