A generalized central limit theorem for critical marked Hawkes processes

Abstract: We prove a central limit type theorem for critical marked Hawkes process. We study the case where the marks are i.i.d. with nonnegative values and their common distribution is either heavy tailed or has finite variance. The kernel function is of a multiplicative form and the mean number of future events triggered by a single event is $1$ (criticality). We also assume that the base intensity function is heavy tailed. We prove convergence in law in the space of tempered distributions of the normalized empirical measure corresponding to the times of events. We also study convergence in law in the Skorokhod space of the normalized event counting process as the time is speeded up. In case when the distribution of marks is heavy tailed, the limit process is a stable process with dependent increments, while in case of finite variance the limit process is the same Gaussian process as for the non marked Hawkes process. We develop a new, robust method that may be applied to other self-exciting systems generalizing Hawkes processes. For example we consider a non marked self-exciting system where the number of excitations caused by single event is heavy tailed.