期刊
ANNALS OF STATISTICS
卷 43, 期 4, 页码 1535-1567出版社
INST MATHEMATICAL STATISTICS
DOI: 10.1214/15-AOS1315
关键词
High-dimensional time series; stochastic regression; vector autoregression; covariance estimation; lasso
资金
- NSA [H98230-10-1-0203]
- NSF [DMS-11-61838, DMS-12-28164]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1545277, 1228164] Funding Source: National Science Foundation
Many scientific and economic problems involve the analysis of high-dimensional time series datasets. However, theoretical studies in high-dimensional statistics to date rely primarily on the assumption of independent and identically distributed (i.i.d.) samples. In this work, we focus on stable Gaussian processes and investigate the theoretical properties of l(1)-regularized estimates in two important statistical problems in the context of high-dimensional time series: (a) stochastic regression with serially correlated errors and (b) transition matrix estimation in vector autoregressive (VAR) models. We derive nonasymptotic upper bounds on the estimation errors of the regularized estimates and establish that consistent estimation under high-dimensional scaling is possible via l(1)-regularization for a large class of stable processes under sparsity constraints. A key technical contribution of the work is to introduce a measure of stability for stationary processes using their spectral properties that provides insight into the effect of dependence on the accuracy of the regularized estimates. With this proposed stability measure, we establish some useful deviation bounds for dependent data, which can be used to study several important regularized estimates in a time series setting.
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