The Multivariate Hawkes Process in High Dimensions: Beyond Mutual Excitation

TL;DR

This paper extends the theoretical understanding of multivariate Hawkes processes by allowing for inhibitory and non-linear interactions, providing new bounds and concentration inequalities for high-dimensional analysis.

Contribution

It introduces a novel approach using thinning and coupling to analyze Hawkes processes beyond mutual excitation and linear link functions.

Findings

Established concentration inequalities for second-order statistics.

Bounded dependence coefficients in high-dimensional Hawkes processes.

Validated theoretical results with simulation studies.

Abstract

The Hawkes process is a class of point processes whose future depends on their own history. Previous theoretical work on the Hawkes process is limited to a special case in which a past event can only increase the occurrence of future events, and the link function is linear. However, in neuronal networks and other real-world applications, inhibitory relationships may be present, and the link function may be non-linear. In this paper, we develop a new approach for investigating the properties of the Hawkes process without the restriction to mutual excitation or linear link functions. To this end, we employ a thinning process representation and a coupling construction to bound the dependence coefficient of the Hawkes process. Using recent developments on weakly dependent sequences, we establish a concentration inequality for second-order statistics of the Hawkes process. We apply this concentration inequality to cross-covariance analysis in the high-dimensional regime, and we verify the theoretical claims with simulation studies.