Lasso Guarantees for Time Series Estimation Under Subgaussian Tails and $β$-Mixing

Many theoretical results on estimation of high dimensional time series require specifying an underlying data generating model (DGM). Instead, this paper relies only on (strict) stationarity and $\beta$-mixing condition to establish consistency of the Lasso when data comes from a $\beta$-mixing process with marginals having subgaussian tails. We establish non-asymptotic inequalities for estimation and prediction errors of the Lasso estimate of the best linear predictor in dependent data. Applications of these results potentially extend to non-Gaussian, non-Markovian and non-linear times series models as the examples we provide demonstrate. In order to prove our results, we derive a novel Hanson-Wright type concentration inequality for $\beta$-mixing subgaussian random vectors that may be of independent interest.