  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.
