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Many theoretical results on estimation of high dimensional time seriesrequire specifying an underlying data generating model (DGM). Instead, alongthe footsteps of~\cite{wong2017lasso}, this paper relies only on (strict)stationarity and $ \beta $-mixing condition to establish consistency of lassowhen data comes from a $\beta$-mixing process with marginals having subgaussiantails. Because of the general assumptions, the data can come from DGMsdifferent than standard time series models such as VAR or ARCH. When the trueDGM is not VAR, the lasso estimates correspond to those of the best linearpredictors using the past observations. We establish non-asymptoticinequalities for estimation and prediction errors of the lasso estimates.Together with~\cite{wong2017lasso}, we provide lasso guarantees that cover fullspectrum of the parameters in specifications of $ \beta $-mixing subgaussiantime series. Applications of these results potentially extend to non-Gaussian,non-Markovian and non-linear times series models as the examples we providedemonstrate. In order to prove our results, we derive a novel Hanson-Wrighttype concentration inequality for $\beta$-mixing subgaussian random vectorsthat may be of independent interest.

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