  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.
