Limit theorems for non linear (compound marked) Hawkes processes
Benjamin Massat
TL;DR
The paper addresses functional limit theorems for non-linear, compound marked Hawkes processes, establishing a functional CLT in $\mathbb{D}([0,1],\mathbb{R})$ along with an explicit convergence-rate bound in the functional 1-Wasserstein distance. The authors fuse Poisson embedding with Malliavin–Stein techniques (Nourdin–Peccati–Réveillac) to handle non-linearity, marks, and lack of monotonicity in the kernel and nonlinear function. The main contributions are the first fCLT for this broad Hawkes class and a quantitative rate $d_W\left(F^{(T)}, \tilde{\sigma}B\right) \le C\frac{\ln(T)}{T^{1/10}}$, with $F^{(T)}_t = \dfrac{L_{tT} - m_{g,1}\int_0^{tT}\lambda_s ds}{\sqrt{T}}$, $m_{g,1}=\int g(x) \vartheta(dx)$, $m_{g,2}=\int |g(x)|^2 \vartheta(dx)$, and $\tilde{\sigma}^2=\sigma^2 m_{g,2}$. This advances the theoretical understanding of non-linear Hawkes dynamics and provides a rigorous tool for assessing asymptotic normality in complex point-process models with marks.
Abstract
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.