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The Hawkes process is a class of point processes whose future depends ontheir own history. Previous theoretical work on the Hawkes process is limitedto a special case in which a past event can only increase the occurrence offuture events, and the link function is linear. However, in neuronal networksand other real-world applications, inhibitory relationships may be present, andthe link function may be non-linear. In this paper, we develop a new approachfor investigating the properties of the Hawkes process without the restrictionto mutual excitation or linear link functions. To this end, we employ athinning process representation and a coupling construction to bound thedependence coefficient of the Hawkes process. Using recent developments onweakly dependent sequences, we establish a concentration inequality forsecond-order statistics of the Hawkes process. We apply this concentrationinequality to cross-covariance analysis in the high-dimensional regime, and weverify the theoretical claims with simulation studies.

The Multivariate Hawkes Process in High Dimensions: Beyond Mutual Excitation