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There has been considerable advance in understanding the properties of sparse regularization procedures in high-dimensional models. In time series context, it is mostly restricted to Gaussian autoregressions or mixing sequences. We study oracle properties of LASSO estimation of weakly sparse vector-autoregressive models with heavy tailed, weakly dependent innovations with virtually no assumption on the conditional heteroskedasticity. In contrast to current literature, our innovation process satisfy an L1 mixingale type condition on the centered conditional covariance matrices. This condition covers L1-NED sequences and strong (ff-) mixing sequences as particular examples. From a modeling perspective, it covers several multivariate-GARCH specifications, such as the BEKK model, and other factor stochastic volatility specifications that were ruled out by assumption in previous studies.