Spectral analysis of multivariate stationary Hawkes processes

TL;DR

This paper proves the validity of frequency-domain inference methods for stationary multivariate Hawkes processes, including consistency and asymptotic normality of the Whittle estimator, and introduces a frequency-domain test for subprocess independence.

Contribution

It establishes theoretical foundations for spectral analysis of Hawkes processes, including new asymptotic results and a simple independence test, applicable under mild conditions.

Findings

Proved consistency of the Whittle estimator for Hawkes processes.

Derived asymptotic normality with explicit covariance.

Demonstrated the methods via simulation studies.

Abstract

We establish the asymptotic validity of frequency-domain inference for stationary multivariate Hawkes processes under mild conditions, bridging the gap between theory and application. By developing upper-bounds on the reduced cumulant measures from the cluster representation of the Hawkes processes, we prove a functional central limit theorem and, as a consequence, consistency of the Whittle estimator under stationarity alone (i.e., the spectral radius of the interactions matrix $\rho(\boldsymbol\nu)<1$), applicable to Hawkes processes with heavy-tailed mutual-excitation kernels. Under mild extra moment conditions, we further obtain asymptotic normality with an explicit limiting covariance in terms of second- and fourth-order cumulant spectral densities. We also propose a simple frequency-domain method to detect joint independence of subprocesses of a multivariate Hawkes process. The performance of the Whittle estimator and the test of independence are demonstrated via simulation studies.