  High-dimensional auto-regressive models provide a natural way to model
influence between $M$ actors given multi-variate time series data for $T$ time
intervals. While there has been considerable work on network estimation, there
is limited work in the context of inference and hypothesis testing. In
particular, prior work on hypothesis testing in time series has been restricted
to linear Gaussian auto-regressive models. From a practical perspective, it is
important to determine suitable statistical tests for connections between
actors that go beyond the Gaussian assumption. In the context of
\emph{high-dimensional} time series models, confidence intervals present
additional estimators since most estimators such as the Lasso and Dantzig
selectors are biased which has led to \emph{de-biased} estimators. In this
paper we address these challenges and provide convergence in distribution
results and confidence intervals for the multi-variate AR(p) model with
sub-Gaussian noise, a generalization of Gaussian noise that broadens
applicability and presents numerous technical challenges. The main technical
challenge lies in the fact that unlike Gaussian random vectors, for
sub-Gaussian vectors zero correlation does not imply independence. The proof
relies on using an intricate truncation argument to develop novel concentration
bounds for quadratic forms of dependent sub-Gaussian random variables. Our
convergence in distribution results hold provided $T = \Omega((s \vee \rho)^2
\log^2 M)$, where $s$ and $\rho$ refer to sparsity parameters which matches
existed results for hypothesis testing with i.i.d. samples. We validate our
theoretical results with simulation results for both block-structured and
chain-structured networks.
