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High dimensional Vector Autoregressions (VAR) have received a lot of interestrecently due to novel applications in health, engineering, finance and thesocial sciences. Three issues arise when analyzing VAR's: (a) The highdimensional nature of the model in the presence of many time series that poseschallenges for consistent estimation of its parameters; (b) the presence oftemporal dependence introduces additional challenges for theoretical analysisof various estimation procedures; (b) the presence of heavy tails in a numberof applications. Recent work, e.g. [Basu and Michailidis, 2015],[Kock andCallot,2015], has addressed consistent estimation of sparse high dimensional,stable Gaussian VAR models based on an $\ell_1$ LASSO procedure. Further, therates obtained are optimal, in the sense that they match those for iid data,plus a multiplicative factor (which is the "price" paid) for temporaldependence.  However, the third issue remains unaddressed in extant literature. This paperextends existing results in the following important direction: it considersconsistent estimation of the parameters of sparse high dimensional VAR modelsdriven by heavy tailed homoscedastic or heteroskedastic noise processes (thatdo not possess all moments). A robust penalized approach (e.g., LASSO) isadopted for which optimal consistency rates and corresponding finite samplebounds for the underlying model parameters are obtain that match those for iiddata, albeit paying a price for temporal dependence. The theoretical resultsare illustrated on VAR models and also on other popular time series models.Notably, the key technical tool used, is a single concentration bound for heavytailed dependent processes.

Robust Estimation of Sparse, High Dimensional Time Series with Polynomial Tails