Title: The Multivariate Hawkes Process in High Dimensions: Beyond Mutual Excitation,
Abstract: The Hawkes process is a class of point processes whose future depends on its
own history. Previous theoretical work on the Hawkes process is limited to the
case of a mutually-exciting process, in which a past event can only increase
the occurrence of future events. However, in neuronal networks and other
real-world applications, inhibitory relationships may be present. In this
paper, we develop a new approach for establishing the properties of the Hawkes
process without the restriction to mutual excitation. 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 in order to establish theoretical results for penalized regression
and clustering analysis in the high-dimensional regime. Our theoretical results
are corroborated by simulation studies and an application to a neuronal spike
train data set.