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：我们证明了霍克斯过程的长期行为完全由子事件的平均数量和离散程度决定。对于亚临界过程，我们在对过程核的极弱条件下建立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程度。对于具有弱离散子事件的临界霍克斯过程，泛函中心极限定理不成立。相反，我们证明了重标度强度过程和重标度霍克斯过程的行为分别类似于无均值回归的CIR过程及其积分过程。我们通过建立重标度霍克斯过程分布与相应极限过程分布之间的Wasserstein距离上界，给出了收敛速率。相比之下，具有强离散子事件的临界霍克斯过程则与亚临界过程共享许多性质，特别是泛函极限定理成立。然而，与亚临界过程不同，具有强离散子事件的临界过程会表现出长程依赖性。