Limit theorems for non linear (compound marked) Hawkes processes

In this article, we fill a gap in the literature on Hawkes processes. In particular, we derive a CLT for a non linear compound marked Hawkes process. We also provide an upper bound on the convergence rate using the functional 1-Wasserstein distance. This result is obtained by discretizing the time line and reducing the problem to the quantification of the distance between finite marginal vectors, as well as between the discretized process and the original one.