Fueled in part by recent applications in neuroscience, high-dimensional Hawkes process have become a popular tool for modeling the network of interactions among multivariate point process data. While evaluating the uncertainty of the network estimates is critical in scientific applications, existing methodological and theoretical work have only focused on estimation. To bridge this gap, this paper proposes a high-dimensional statistical inference procedure with theoretical guarantees for multivariate Hawkes process. Key to this inference procedure is a new concentration inequality on the first- and second-order statistics for integrated stochastic processes, which summarizes the entire history of the process. We apply this concentration inequality, combining a recent result on martingale central limit theory, to give an upper bounds for the convergence rate of the test statistics. We verify our theoretical results with extensive simulation and an application to a neuron spike train data set.


Biostatistics | Multivariate Analysis | Probability | Statistical Methodology | Statistical Theory