The subject of this paper is autoregressive (AR) modeling of a stationary, Gaussian discrete time process, based on a finite sequence of observations. The process is assumed to admit an AR($\infty$) representation with exponentially decaying coefficients. We adopt the nonparametric minimax framework and study how well the process can be approximated by a finiteorder AR model. A lower bound on the accuracy of AR approximations is derived, and a nonasymptotic upper bound on the accuracy of the regularized least squares estimator is established. It is shown that with a “proper” choice of the model order, this estimator is minimax optimal in order. These considerations lead also to a nonasymptotic upper bound on the mean squared error of the associated onestep predictor. A numerical study compares the common model selection procedures to the minimax optimal order choice.
"Nonasymptotic bounds for autoregressive time series modeling." Ann. Statist. 29 (2) 417 - 444, April 2001. https://doi.org/10.1214/aos/1009210547