  Vector autoregressive models characterize a variety of time series in which
linear combinations of current and past observations can be used to accurately
predict future observations. For instance, each element of an observation
vector could correspond to a different node in a network, and the parameters of
an autoregressive model would correspond to the impact of the network structure
on the time series evolution. Often these models are used successfully in
practice to learn the structure of social, epidemiological, financial, or
biological neural networks. However, little is known about statistical
guarantees on estimates of such models in non-Gaussian settings. This paper
addresses the inference of the autoregressive parameters and associated network
structure within a generalized linear model framework that includes Poisson and
Bernoulli autoregressive processes. At the heart of this analysis is a
sparsity-regularized maximum likelihood estimator. While
sparsity-regularization is well-studied in the statistics and machine learning
communities, those analysis methods cannot be applied to autoregressive
generalized linear models because of the correlations and potential
heteroscedasticity inherent in the observations. Sample complexity bounds are
derived using a combination of martingale concentration inequalities and modern
empirical process techniques for dependent random variables. These bounds,
which are supported by several simulation studies, characterize the impact of
various network parameters on estimator performance.
