  Fueled in part by recent applications in neuroscience, the multivariate
Hawkes process has become a popular tool for modeling the network of
interactions among high-dimensional point process data. While evaluating the
uncertainty of the network estimates is critical in scientific applications,
existing methodological and theoretical work has primarily addressed
estimation. To bridge this gap, this paper develops a new statistical inference
procedure for high-dimensional Hawkes processes. The key ingredient for this
inference procedure is a new concentration inequality on the first- and
second-order statistics for integrated stochastic processes, which summarize
the entire history of the process. Combining recent results on martingale
central limit theory with the new concentration inequality, we then
characterize the convergence rate of the test statistics. We illustrate finite
sample validity of our inferential tools via extensive simulations and
demonstrate their utility by applying them to a neuron spike train data set.
