Covariance and precision matrix estimation for high-dimensional time series

Xiaohui Chen, Mengyu Xu, Wei Biao Wu

Research output: Contribution to journalArticlepeer-review


We consider estimation of covariance matrices and their inverses (a.k.a. precision matrices) for high-dimensional stationary and locally stationary time series. In the latter case the covariance matrices evolve smoothly in time, thus forming a covariance matrix function. Using the functional dependence measure of Wu [Proc. Natl. Acad. Sci. USA 102 (2005) 14150-14154 (electronic)], we obtain the rate of convergence for the thresholded estimate and illustrate how the dependence affects the rate of convergence. Asymptotic properties are also obtained for the precision matrix estimate which is based on the graphical Lasso principle. Our theory substantially generalizes earlier ones by allowing dependence, by allowing nonstationarity and by relaxing the associated moment conditions.

Original languageEnglish (US)
Pages (from-to)2994-3021
Number of pages28
JournalAnnals of Statistics
Issue number6
StatePublished - Dec 2013


  • Consistency
  • Covariance matrix
  • Dependence
  • Functional dependence measure
  • High-dimensional inference
  • Lasso
  • Nagaev inequality
  • Nonstationary time series
  • Precision matrix
  • Sparsity
  • Spatial-temporal processes
  • Thresholding

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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