Functional limit theorems for quasi-stationary Hawkes processes
In this talk, we introduce several functional law of large numbers (FLLN) and functional central limit theorem (FCLT) for quasi-stationary Hawkes processes. Under some divergence conditions on triggered events, We prove that the normalized point processes can be approximated in distribution by a long-range dependent Gaussian process. Differently, both FLLN and FCLT fail when triggered events satisfy aggregation conditions. In this case, we prove that the rescale Hawkes process converges weakly to the integral of a critical branching diffusion with immigration. Also, the convergence rate is deduced in terms of Fourier-Laplace distance bound and Wasserstein distance bound. This talk is based on a joint work with Ulrich Horst.