We develop concentration inequalities for sup norm of a vector linear processes on mixingale sequences with sub-Weibull tails. These inequalities make use of the Beveridge-Nelson decomposition, which reduces the problem to concentration for sup-norm of a vector-mixingale or its weighted sum. This inequality is used in the derivation of concentration bound for the maximum entrywise norm of auto-covariance matrices of linear processes. These results are useful for estimation bounds for high dimensional vector-autoregressive processes estimated using L1 regularization, high-dimensional Gaussian bootstrap for time-series, and long-run covariance matrix estimation.