Normal approximation of Functionals of Point Processes: Application to Hawkes Processes

TL;DR

This paper develops explicit bounds for how closely functionals of point processes, including Hawkes processes, approximate Gaussian distributions, using advanced probabilistic techniques.

Contribution

It introduces a novel approach combining Stein's method, Malliavin calculus, and Poisson embedding to analyze convergence rates of Hawkes processes to Gaussian limits.

Findings

Explicit Wasserstein distance bounds for point process functionals

Convergence rates for stable continuous Hawkes processes

Bounds for nearly unstable Hawkes processes

Abstract

In this paper, we derive an explicit upper bound for the Wasserstein distance between a functional of point processes and a Gaussian distribution. Using Stein's method in conjunction with Malliavin's calculus and the Poisson embedding representation, our result applies to a variety of point processes including discrete and continuous Hawkes processes. In particular, we establish an explicit convergence rate for stable continuous non-linear Hawkes processes and for discrete Hawkes processes. Finally, we obtain an upper bound in the context of nearly unstable Hawkes processes.