Lasso guarantees for $\beta$-mixing heavy-tailed time series

Abstract

Many theoretical results for lasso require the samples to be i.i.d. Recent work has provided guarantees for lasso assuming that the time series is generated by a sparse Vector Autoregressive (VAR) model with Gaussian innovations. Proofs of these results rely critically on the fact that the true data generating mechanism (DGM) is a finite-order Gaussian VAR. This assumption is quite brittle: linear transformations, including selecting a subset of variables, can lead to the violation of this assumption. In order to break free from such assumptions, we derive nonasymptotic inequalities for estimation error and prediction error of lasso estimate of the best linear predictor without assuming any special parametric form of the DGM. Instead, we rely only on (strict) stationarity and geometrically decaying $\beta$-mixing coefficients to establish error bounds for lasso for sub-Weibull random vectors. The class of sub-Weibull random variables that we introduce includes sub-Gaussian and subexponential random variables but also includes random variables with tails heavier than an exponential. We also show that, for Gaussian processes, the $\beta$-mixing condition can be relaxed to summability of the $\alpha$-mixing coefficients. Our work provides an alternative proof of the consistency of lasso for sparse Gaussian VAR models. But the applicability of our results extends to non-Gaussian and nonlinear times series models as the examples we provide demonstrate.